 7.7.1.1: A rectangular array of real numbers that can be used to solve a sys...
 7.7.4.1: Both and represent the ________ of the matrix A
 7.7.2.1: Two matrices are ________ when their corresponding entries are equal.
 7.7.5.1: The method of using determinants to solve a system of linear equati...
 7.7.3.1: In a ________ matrix, the number of rows equals the number of columns
 7.1: In Exercises 14, determine the order of the matrix. 405
 7.7.1.2: A matrix is ________ when the number of rows equals the number of c...
 7.7.4.2: The ________ of the entry is the determinant of the matrix obtained...
 7.7.2.2: When performing matrix operations, real numbers are often referred ...
 7.7.5.2: Three points are ________ when the points lie on the same line.
 7.7.3.2: If there exists an matrix such that then is called the ________ of
 7.2: In Exercises 14, determine the order of the matrix. 32170164
 7.7.1.3: For a square matrix, the entries are the ________ ________ entries.
 7.7.4.3: The ________ of the entry of the square matrix is given by Mij a A ...
 7.7.2.3: A matrix consisting entirely of zeros is called a ________ matrix a...
 7.7.5.3: The area of a triangle with vertices and is given by ________.
 7.7.3.3: 3. If a matrix has an inverse, then it is called invertible or ____...
 7.3: In Exercises 14, determine the order of the matrix. 3
 7.7.1.4: A matrix with only one row is called a ________ matrix, and a matri...
 7.7.4.4: The method of finding the determinant of a matrix of order or great...
 7.7.2.4: The matrix consisting of 1s on its main diagonal and 0s elsewhere i...
 7.7.5.4: A message written according to a secret code is called a ________.
 7.7.3.4: If is an invertible matrix, then the system of linear equations rep...
 7.4: In Exercises 14, determine the order of the matrix. 6 2 580
 7.7.1.5: The matrix derived from a system of linear equations is called the ...
 7.7.4.5: In Exercises 522, find the determinant of the matrix. 4
 7.7.2.5: In Exercises 58, find x and y x72y 47222
 7.7.5.5: To encode a message, choose an invertible matrix and multiply the _...
 7.7.3.5: In Exercises 512, show that is the inverse of A A 2513, B 3512 A 2
 7.5: In Exercises 5 and 6, write the augmented matrix for the system of ...
 7.7.1.6: The matrix derived from the coefficients of a system of linear equa...
 7.7.4.6: In Exercises 522, find the determinant of the matrix. 10
 7.7.2.6: In Exercises 58, find x and y 5yx8 512138
 7.7.5.6: If a message is encoded using an invertible matrix then the message...
 7.7.3.6: In Exercises 512, show that is the inverse of A A 1112,B B2111 A
 7.6: In Exercises 5 and 6, write the augmented matrix for the system of ...
 7.7.1.7: Two matrices are called ________ when one of the matrices can be ob...
 7.7.4.7: In Exercises 522, find the determinant of the matrix. 8243
 7.7.2.7: In Exercises 58, find x and y 163041325154460 163041322x 1153y 543x0
 7.7.5.7: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.7: In Exercises 512, show that is the inverse of A A 1324, B 124321 A 1
 7.7: In Exercises 7 and 8, write the system of linear equations represen...
 7.7.1.8: A matrix in rowechelon form is in ________ ________ ________ when ...
 7.7.4.8: In Exercises 522, find the determinant of the matrix. 9602 8
 7.7.2.8: In Exercises 58, find x and y x 21782y232xy 2 2x 61781823811
 7.7.5.8: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.8: In Exercises 512, show that is the inverse of A A 1213,B B 153211A
 7.8: In Exercises 7 and 8, write the system of linear equations represen...
 7.7.1.9: In Exercises 916, determine the order of the matrix. 7 0
 7.7.4.9: In Exercises 522, find the determinant of the matrix. 6523
 7.7.2.9: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.9: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.9: In Exercises 512, show that is the inverse of A A 210171131172, B12...
 7.9: In Exercises 9 and 10, write the matrix in rowechelon form. (Remem...
 7.7.1.10: In Exercises 916, determine the order of the matrix. 5 387
 7.7.4.10: In Exercises 522, find the determinant of the matrix. 3438
 7.7.2.10: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.10: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.10: In Exercises 512, show that is the inverse of A A410121541, B 14211...
 7.10: In Exercises 9 and 10, write the matrix in rowechelon form. (Remem...
 7.7.1.11: In Exercises 916, determine the order of the matrix. 2363
 7.7.4.11: In Exercises 522, find the determinant of the matrix. 7300
 7.7.2.11: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.11: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.11: In Exercises 512, show that is the inverse of A A2313001120211110, ...
 7.11: In Exercises 1114, write the system of linear equations represented...
 7.7.1.12: In Exercises 916, determine the order of the matrix. 3017011536037 2
 7.7.4.12: In Exercises 522, find the determinant of the matrix. 4030
 7.7.2.12: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.12: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.12: In Exercises 512, show that is the inverse of A A1110111101211001, ...
 7.12: In Exercises 1114, write the system of linear equations represented...
 7.7.1.13: In Exercises 916, determine the order of the matrix. 3394520
 7.7.4.13: In Exercises 522, find the determinant of the matrix. 2063
 7.7.2.13: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.13: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.13: In Exercises 1324, find the inverse of the matrix (if it exists). 2003
 7.13: In Exercises 1114, write the system of linear equations represented...
 7.7.1.14: In Exercises 916, determine the order of the matrix. 706541 3
 7.7.4.14: In Exercises 522, find the determinant of the matrix. 2639
 7.7.2.14: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.14: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.14: In Exercises 1324, find the inverse of the matrix (if it exists). 1327
 7.14: In Exercises 1114, write the system of linear equations represented...
 7.7.1.15: In Exercises 916, determine the order of the matrix. 183609139
 7.7.4.15: In Exercises 522, find the determinant of the matrix. 3621
 7.7.2.15: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.15: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.15: In Exercises 1324, find the inverse of the matrix (if it exists). 1223
 7.15: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.16: In Exercises 916, determine the order of the matrix. 345119 1
 7.7.4.16: In Exercises 522, find the determinant of the matrix. 4275
 7.7.2.16: In Exercises 916, if possible, find (a) (b) (c) and (d A B, A B, 3A...
 7.7.5.16: In Exercises 716, use Cramers Rule to solve (if possible) the syste...
 7.7.3.16: In Exercises 1324, find the inverse of the matrix (if it exists). 7...
 7.16: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.17: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.17: In Exercises 522, find the determinant of the matrix. 2371
 7.7.2.17: In Exercises 1722, evaluate the expression 53067211101486A B
 7.7.5.17: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.17: In Exercises 1324, find the inverse of the matrix (if it exists). 3412
 7.17: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.18: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.18: In Exercises 522, find the determinant of the matrix. 2451
 7.7.2.18: In Exercises 1722, evaluate the expression 6180035111271
 7.7.5.18: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.18: In Exercises 1324, find the inverse of the matrix (if it exists). 4...
 7.18: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.19: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.19: In Exercises 522, find the determinant of the matrix. 71263
 7.7.2.19: In Exercises 1722, evaluate the expression 400213231620
 7.7.5.19: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.19: In Exercises 1324, find the inverse of the matrix (if it exists). 1...
 7.19: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.20: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.20: In Exercises 522, find the determinant of the matrix. 0362
 7.7.2.20: In Exercises 1722, evaluate the expression 125 24014 6 18 94
 7.7.5.20: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.20: In Exercises 1324, find the inverse of the matrix (if it exists). 3...
 7.20: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.21: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.21: In Exercises 522, find the determinant of the matrix. 1261313
 7.7.2.21: In Exercises 1722, evaluate the expression 307326831247491
 7.7.5.21: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.21: In Exercises 1324, find the inverse of the matrix (if it exists). 5...
 7.21: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.22: In Exercises 1722, write the augmented matrix for the system of lin...
 7.7.4.22: In Exercises 522, find the determinant of the matrix. 231431
 7.7.2.22: In Exercises 1722, evaluate the expression 142911131653014137965113
 7.7.5.22: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.22: In Exercises 1324, find the inverse of the matrix (if it exists). 1...
 7.22: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.23: In Exercises 2328, write the system of linear equations represented...
 7.7.4.23: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.23: In Exercises 2326, use the matrix capabilities of a graphing utilit...
 7.7.5.23: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.23: In Exercises 1324, find the inverse of the matrix (if it exists). 8...
 7.23: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.24: In Exercises 2328, write the system of linear equations represented...
 7.7.4.24: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.24: In Exercises 2326, use the matrix capabilities of a graphing utilit...
 7.7.5.24: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.24: In Exercises 1324, find the inverse of the matrix (if it exists). 0...
 7.24: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.25: In Exercises 2328, write the system of linear equations represented...
 7.7.4.25: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.25: In Exercises 2326, use the matrix capabilities of a graphing utilit...
 7.7.5.25: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.25: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.25: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.26: In Exercises 2328, write the system of linear equations represented...
 7.7.4.26: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.26: In Exercises 2326, use the matrix capabilities of a graphing utilit...
 7.7.5.26: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.26: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.26: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.27: In Exercises 2328, write the system of linear equations represented...
 7.7.4.27: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.27: In Exercises 2734, solve for in the equation, where X 2A 2B
 7.7.5.27: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.27: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.27: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.28: In Exercises 2328, write the system of linear equations represented...
 7.7.4.28: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.28: In Exercises 2734, solve for in the equation, where X 3A 2B
 7.7.5.28: In Exercises 1728, use a determinant to find the area with the give...
 7.7.3.28: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.28: In Exercises 1528, use matrices to solve the system of equations (i...
 7.7.1.29: In Exercises 2932, identify the elementary row operation(s) being p...
 7.7.4.29: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.29: In Exercises 2734, solve for in the equation, where 2X A B
 7.7.5.29: In Exercises 29 and 30, find a value of such that the triangle with...
 7.7.3.29: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.29: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.30: In Exercises 2932, identify the elementary row operation(s) being p...
 7.7.4.30: In Exercises 2330, use the matrix capabilities of a graphing utilit...
 7.7.2.30: In Exercises 2734, solve for in the equation, where 2X 2A B
 7.7.5.30: In Exercises 29 and 30, find a value of such that the triangle with...
 7.7.3.30: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.30: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.31: In Exercises 2932, identify the elementary row operation(s) being p...
 7.7.4.31: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.31: In Exercises 2734, solve for in the equation, where 2X 3A B
 7.7.5.31: In Exercises 31 and 32, find a value of such that the triangle with...
 7.7.3.31: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.31: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.32: In Exercises 2932, identify the elementary row operation(s) being p...
 7.7.4.32: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.32: In Exercises 2734, solve for in the equation, where 3X 4A 2B
 7.7.5.32: In Exercises 31 and 32, find a value of such that the triangle with...
 7.7.3.32: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.32: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.33: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.33: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.33: In Exercises 2734, solve for in the equation, where 2A 4B 2X 3
 7.7.5.33: A large region of forest has been infested with gypsy moths. The re...
 7.7.3.33: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.33: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.34: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.34: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.34: In Exercises 2734, solve for in the equation, where 5A 6B 3X2
 7.7.5.34: Botany A botanist is studying the plants growing in a triangular tr...
 7.7.3.34: In Exercises 2534, use the matrix capabilities of a graphing utilit...
 7.34: In Exercises 2934, use matrices to solve the system of equations (i...
 7.7.1.35: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.35: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.35: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.35: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.35: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.35: In Exercises 35 and 36, use the matrix capabilities of a graphing u...
 7.7.1.36: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.36: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.36: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.36: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.36: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.36: In Exercises 35 and 36, use the matrix capabilities of a graphing u...
 7.7.1.37: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.37: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.37: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.37: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.37: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.37: In Exercises 3740, find x and y 1yx917129x
 7.7.1.38: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.38: In Exercises 3138, find all the (a) minors and (b) cofactors of the...
 7.7.2.38: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.38: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.38: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.38: In Exercises 3740, find x and y 1x405y184050
 7.7.1.39: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.39: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.39: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.39: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.39: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.39: In Exercises 3740, find x and y x 30243y 54y26x5x 102431644261
 7.7.1.40: In Exercises 3340, fill in the blank(s) using elementary row operat...
 7.7.4.40: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.40: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.40: In Exercises 3540, use a determinant to determine whether the point...
 7.7.3.40: In Exercises 3540, use the formula on page 527 to find the inverse ...
 7.40: In Exercises 3740, find x and y 906431271540x 9012x431x 107152y0
 7.7.1.41: In Exercises 41 and 42, (a) perform the row operations to solve the...
 7.7.4.41: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.41: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.41: In Exercises 41 and 42, find such that the points are collinear. 2,...
 7.7.3.41: In Exercises 4144, use the inverse matrix found in Exercise 15 to s...
 7.41: In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, ...
 7.7.1.42: In Exercises 41 and 42, (a) perform the row operations to solve the...
 7.7.4.42: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.42: In Exercises 3542, if possible, find and state the order of the res...
 7.7.5.42: In Exercises 41 and 42, find such that the points are collinear. 6,...
 7.7.3.42: In Exercises 4144, use the inverse matrix found in Exercise 15 to s...
 7.42: In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, ...
 7.7.1.43: In Exercises 4346, determine whether the matrix is in rowechelon f...
 7.7.4.43: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.43: In Exercises 4346, use the matrix capabilities of a graphing utilit...
 7.7.5.43: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.43: In Exercises 4144, use the inverse matrix found in Exercise 15 to s...
 7.43: In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, ...
 7.7.1.44: In Exercises 4346, determine whether the matrix is in rowechelon f...
 7.7.4.44: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.44: In Exercises 4346, use the matrix capabilities of a graphing utilit...
 7.7.5.44: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.44: In Exercises 4144, use the inverse matrix found in Exercise 15 to s...
 7.44: In Exercises 4144, if possible, find (a) (b) (c) and (d) A B, A B, ...
 7.7.1.45: In Exercises 4346, determine whether the matrix is in rowechelon f...
 7.7.4.45: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.45: In Exercises 4346, use the matrix capabilities of a graphing utilit...
 7.7.5.45: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.45: In Exercises 45 and 46, use the inverse matrix found in Exercise 19...
 7.45: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.46: In Exercises 4346, determine whether the matrix is in rowechelon f...
 7.7.4.46: In Exercises 3946, find the determinant of the matrix. Expand by co...
 7.7.2.46: In Exercises 4346, use the matrix capabilities of a graphing utilit...
 7.7.5.46: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.46: In Exercises 45 and 46, use the inverse matrix found in Exercise 19...
 7.46: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.47: In Exercises 4750, write the matrix in rowechelon form. (Remember ...
 7.7.4.47: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.47: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A...
 7.7.5.47: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.47: In Exercises 47 and 48, use the inverse matrix found in Exercise 34...
 7.47: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.48: In Exercises 4750, write the matrix in rowechelon form. (Remember ...
 7.7.4.48: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.48: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A...
 7.7.5.48: In Exercises 4348, use a determinant to find an equation of the lin...
 7.7.3.48: In Exercises 47 and 48, use the inverse matrix found in Exercise 34...
 7.48: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.49: In Exercises 4750, write the matrix in rowechelon form. (Remember ...
 7.7.4.49: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.49: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, .A ...
 7.7.5.49: In Exercises 49 and 50, (a) write the uncoded row matrices for the ...
 7.7.3.49: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.49: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.50: In Exercises 4750, write the matrix in rowechelon form. (Remember ...
 7.7.4.50: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.50: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A...
 7.7.5.50: In Exercises 49 and 50, (a) write the uncoded row matrices for the ...
 7.7.3.50: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.50: In Exercises 4550, evaluate the expression. If it is not possible, ...
 7.7.1.51: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.51: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.51: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A...
 7.7.5.51: In Exercises 51 and 52, (a) write the uncoded row matrices for the ...
 7.7.3.51: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.51: In Exercises 5154, solve for in the equation, where X 2A 3B
 7.7.1.52: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.52: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.52: In Exercises 4752, if possible, find (a) (b) and (c) A2 AB, BA, . A...
 7.7.5.52: In Exercises 51 and 52, (a) write the uncoded row matrices for the ...
 7.7.3.52: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.52: In Exercises 5154, solve for in the equation, where 6X 4A 3B
 7.7.1.53: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.53: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.53: In Exercises 5356, evaluate the expression. Use the matrix capabili...
 7.7.5.53: In Exercises 5356, write a cryptogram for the message using the mat...
 7.7.3.53: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.53: In Exercises 5154, solve for in the equation, where 3X 2A B
 7.7.1.54: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.54: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.54: In Exercises 5356, evaluate the expression. Use the matrix capabili...
 7.7.5.54: In Exercises 5356, write a cryptogram for the message using the mat...
 7.7.3.54: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.54: In Exercises 5154, solve for in the equation, where 2A 5B 3X
 7.7.1.55: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.55: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.55: In Exercises 5356, evaluate the expression. Use the matrix capabili...
 7.7.5.55: In Exercises 5356, write a cryptogram for the message using the mat...
 7.7.3.55: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.55: In Exercises 5558, find if possible. A 2325,A B 312108 A 23
 7.7.1.56: In Exercises 5156, use the matrix capabilities of a graphing utilit...
 7.7.4.56: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.56: In Exercises 5356, evaluate the expression. Use the matrix capabili...
 7.7.5.56: In Exercises 5356, write a cryptogram for the message using the mat...
 7.7.3.56: In Exercises 4956, use an inverse matrix to solve (if possible) the...
 7.56: In Exercises 5558, find if possible. A 5711422 B 42015124030A
 7.7.1.57: In Exercises 5760, write the system of linear equations represented...
 7.7.4.57: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.57: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.57: In Exercises 5760, use to decode the cryptogram. A1325 11 21 64 112...
 7.7.3.57: In Exercises 5760, use the matrix capabilities of a graphing utilit...
 7.57: In Exercises 5558, find if possible. A 5711422, B 4201240
 7.7.1.58: In Exercises 5760, write the system of linear equations represented...
 7.7.4.58: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.58: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.58: In Exercises 5760, use to decode the cryptogram. A2334A 85 120 6 8 ...
 7.7.3.58: In Exercises 5760, use the matrix capabilities of a graphing utilit...
 7.58: In Exercises 5558, find if possible. A 6 5 7, B 148
 7.7.1.59: In Exercises 5760, write the system of linear equations represented...
 7.7.4.59: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.59: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.59: In Exercises 5760, use to decode the cryptogram. A 116102013A 1 9 1...
 7.7.3.59: In Exercises 5760, use the matrix capabilities of a graphing utilit...
 7.59: In Exercises 5962, evaluate the expression. If it is not possible, ...
 7.7.1.60: In Exercises 5760, write the system of linear equations represented...
 7.7.4.60: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.60: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.60: In Exercises 5760, use to decode the cryptogram. A3044252138 112 83...
 7.7.3.60: In Exercises 5760, use the matrix capabilities of a graphing utilit...
 7.60: In Exercises 5962, evaluate the expression. If it is not possible, ...
 7.61: In Exercises 5962, evaluate the expression. If it is not possible, ...
 7.7.1.61: In Exercises 6164, an augmented matrix that represents a system of ...
 7.7.4.61: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.61: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.61: In Exercises 61 and 62, decode the cryptogram by using the inverse ...
 7.7.3.61: In Exercises 6164, you invest in AAArated bonds, Arated bonds, an...
 7.7.3.62: In Exercises 6164, you invest in AAArated bonds, Arated bonds, an...
 7.62: In Exercises 5962, evaluate the expression. If it is not possible, ...
 7.7.1.62: In Exercises 6164, an augmented matrix that represents a system of ...
 7.7.4.62: In Exercises 4762, find the determinant of the matrix. Expand by co...
 7.7.2.62: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.62: In Exercises 61 and 62, decode the cryptogram by using the inverse ...
 7.7.3.63: In Exercises 6164, you invest in AAArated bonds, Arated bonds, an...
 7.63: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.1.63: In Exercises 6164, an augmented matrix that represents a system of ...
 7.7.4.63: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.2.63: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.63: Decoding a Message The following cryptogram was encoded with a matr...
 7.7.3.64: In Exercises 6164, you invest in AAArated bonds, Arated bonds, an...
 7.64: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.1.64: In Exercises 6164, an augmented matrix that represents a system of ...
 7.7.4.64: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.2.64: In Exercises 5764, (a) write the system of linear equations as a ma...
 7.7.5.64: Decoding a Message The following cryptogram was encoded with a matr...
 7.7.3.65: In Exercises 6568, consider the circuit shown in the figure. The cu...
 7.65: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.1.65: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.65: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.2.65: Manufacturing A corporation has four factories, each of which manuf...
 7.7.5.65: Circuit Analysis Consider the circuit in the figure. The currents a...
 7.7.3.66: In Exercises 6568, consider the circuit shown in the figure. The cu...
 7.66: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.1.66: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.66: In Exercises 6366, use the matrix capabilities of a graphing utilit...
 7.7.2.66: Vacation Packages A vacation service has identified four resort hot...
 7.7.5.66: . Pulley System A system of pulleys that is assumed frictionless an...
 7.7.3.67: In Exercises 6568, consider the circuit shown in the figure. The cu...
 7.67: Manufacturing A tire corporation has three factories, each of which...
 7.7.1.67: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.67: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 1003, B ...
 7.7.2.67: Agriculture A fruit grower raised two crops, apples and peaches. Ea...
 7.7.5.67: In Exercises 6770, determine whether the statement is true or false...
 7.7.3.68: In Exercises 6568, consider the circuit shown in the figure. The cu...
 7.68: Manufacturing A power tool company has four manufacturing plants, e...
 7.7.1.68: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.68: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 2412,B B...
 7.7.2.68: Inventory A company sells five models of computers through three re...
 7.7.5.68: In Exercises 6770, determine whether the statement is true or false...
 7.7.3.69: In Exercises 69 and 70, find the numbers of bags of potting soil th...
 7.69: Manufacturing An electronics manufacturing company produces three d...
 7.7.1.69: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.69: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 4302, B ...
 7.7.2.69: Revenue An electronics manufacturer produces three models of LCD te...
 7.7.5.69: In Exercises 6770, determine whether the statement is true or false...
 7.7.3.70: In Exercises 69 and 70, find the numbers of bags of potting soil th...
 7.70: Cell Phone Charges The payasyougo charges (in dollars per minute...
 7.7.1.70: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.70: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 5341, B ...
 7.7.2.70: Labor/Wage Requirements A company that manufactures boats has the f...
 7.7.5.70: In Exercises 6770, determine whether the statement is true or false...
 7.7.3.71: Floral Design A florist is creating 10 centerpieces for the tables ...
 7.71: In Exercises 7174, show that is the inverse of A A 4712,B B 2714 A 47
 7.7.1.71: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.71: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 03012421...
 7.7.2.71: Profit At a local dairy mart, the numbers of gallons of skim milk, ...
 7.7.5.71: Writing Use your schools library, the Internet, or some other refer...
 7.7.3.72: International Travel The table shows the numbers of international t...
 7.72: In Exercises 7174, show that is the inverse of A A 51112,B B 21115 A
 7.7.1.72: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.72: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 31223004...
 7.7.2.72: Voting Preferences The matrix From RD I is called a stochastic matr...
 7.7.5.72: Writing (a) State Cramers Rule for solving a system of linear equat...
 7.7.3.73: In Exercises 73 and 74, determine whether the statement is true or ...
 7.73: In Exercises 7174, show that is the inverse of A A 116102013 B 2323...
 7.7.1.73: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.73: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 11020111...
 7.7.2.73: Voting Preferences Use a graphing utility to find and for the matri...
 7.7.5.73: 3. Reasoning Use determinants to find the area of a triangle with v...
 7.7.3.74: In Exercises 73 and 74, determine whether the statement is true or ...
 7.74: In Exercises 7174, show that is the inverse of A A 118104012,B B 23...
 7.7.1.74: In Exercises 6574, use matrices to solve the system of equations (i...
 7.7.4.74: In Exercises 6774, find (a) (b) (c) and (d) AB, B, A, AB A 21301112...
 7.7.2.74: The numbers of calories burned by individuals of different body wei...
 7.7.5.74: HOW DO YOU SEE IT? At this point in the text, you have learned seve...
 7.7.3.75: Writing Explain how to determine whether the inverse of a matrix ex...
 7.75: In Exercises 7578, find the inverse of the matrix (if it exists). 6...
 7.7.1.75: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.75: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.75: In Exercises 75 and 76, determine whether the statement is true or ...
 7.7.3.76: Writing Explain in your own words how to write a system of three li...
 7.76: In Exercises 7578, find the inverse of the matrix (if it exists). 3253
 7.7.1.76: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.76: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.76: In Exercises 75 and 76, determine whether the statement is true or ...
 7.7.3.77: Conjecture Consider matrices of the form (a) Write a matrix and a m...
 7.77: In Exercises 7578, find the inverse of the matrix (if it exists). 2...
 7.7.1.77: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.77: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.77: In Exercises 7780, use the matrices Show that A B2 A2 2AB B2.
 7.7.3.78: HOW DO YOU SEE IT? Let be the matrix Use the determinant of to stat...
 7.78: In Exercises 7578, find the inverse of the matrix (if it exists). 0...
 7.7.1.78: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.78: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.78: In Exercises 7780, use the matrices Show that A B2 A2 2AB B2.
 7.79: In Exercises 7982, use the matrix capabilities of a graphing utilit...
 7.7.1.79: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.79: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.79: In Exercises 7780, use the matrices Show that A BA BA2 B2.
 7.80: In Exercises 7982, use the matrix capabilities of a graphing utilit...
 7.7.1.80: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.80: In Exercises 7580, evaluate the determinant(s) to verify the equati...
 7.7.2.80: In Exercises 7780, use the matrices Show that A B2 A2 AB BA B2.
 7.81: In Exercises 7982, use the matrix capabilities of a graphing utilit...
 7.7.1.81: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.81: In Exercises 8188, solve for x x12x2
 7.7.2.81: Think About It If and are real numbers such that and then However, ...
 7.82: In Exercises 7982, use the matrix capabilities of a graphing utilit...
 7.7.1.82: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.82: In Exercises 8188, solve for x x14x20
 7.7.2.82: Think About It If and are real numbers such that then or However, i...
 7.83: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.83: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.83: In Exercises 8188, solve for x x21x 2 1
 7.7.2.83: Conjecture Let and be unequal diagonal matrices of the same order. ...
 7.84: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.84: In Exercises 7584, use matrices to solve the system of equations (i...
 7.7.4.84: In Exercises 8188, solve for x x 112x4
 7.7.2.84: Matrices with Complex Entries Let and let and (a) Find and Identify...
 7.85: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.85: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.85: In Exercises 8188, solve for x x 13 2x 20
 7.7.2.85: Finding Matrices Find two matrices and such that AB BA.
 7.86: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.86: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.86: In Exercises 8188, solve for x x 23 1x0
 7.7.2.86: HOW DO YOU SEE IT? A corporation has three factories, each of which...
 7.87: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.87: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.87: In Exercises 8188, solve for x x 312x 20
 7.88: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.88: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.88: In Exercises 8188, solve for x x 472x 50
 7.89: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.89: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.89: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.90: In Exercises 8390, use the formula on page 527 to find the inverse ...
 7.7.1.90: In Exercises 8590, use the matrix capabilities of a graphing utilit...
 7.7.4.90: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.91: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.91: In Exercises 9194, determine whether the two systems of linear equa...
 7.7.4.91: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.92: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.92: In Exercises 9194, determine whether the two systems of linear equa...
 7.7.4.92: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.93: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.93: In Exercises 9194, determine whether the two systems of linear equa...
 7.7.4.93: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.94: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.94: In Exercises 9194, determine whether the two systems of linear equa...
 7.7.4.94: In Exercises 8994, evaluate the determinant in which the entries ar...
 7.95: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.95: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.95: In Exercises 95 and 96, determine whether the statement is true or ...
 7.96: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.96: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.96: In Exercises 95 and 96, determine whether the statement is true or ...
 7.97: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.97: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.97: Providing a Counterexample Find square matrices and to demonstrate ...
 7.98: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.98: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.98: Conjecture Consider square matrices in which the entries are consec...
 7.99: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.99: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.99: Writing Write a brief paragraph explaining the difference between a...
 7.100: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.100: In Exercises 95100, use a system of equations to find the quadratic...
 7.7.4.100: Think About It Let be a matrix such that Is it possible to find Exp...
 7.101: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.101: From 2000 through 2011, the numbers of new cases of a waterborne di...
 7.7.4.101: In Exercises 101103, a property of determinants is given ( and are ...
 7.102: In Exercises 91102, use an inverse matrix to solve (if possible) th...
 7.7.1.102: Breeding Facility A city zoo borrowed $2,000,000 at simple annual i...
 7.7.4.102: In Exercises 101103, a property of determinants is given ( and are ...
 7.103: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.103: Museum A natural history museum borrowed $2,000,000 at simple annua...
 7.7.4.103: In Exercises 101103, a property of determinants is given ( and are ...
 7.104: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.104: Mathematical Modeling A video of the path of a ball thrown by a bas...
 7.7.4.104: HOW DO YOU SEE IT? Explain why the determinant of the matrix is equ...
 7.105: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.105: In Exercises 105 and 106, determine whether the statement is true o...
 7.7.4.105: Conjecture A diagonal matrix is a square matrix with all zero entri...
 7.106: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.106: In Exercises 105 and 106, determine whether the statement is true o...
 7.107: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.107: Think About It What is the relationship between the three elementar...
 7.108: In Exercises 103108, use the matrix capabilities of a graphing util...
 7.7.1.108: HOW DO YOU SEE IT? Determine whether the matrix below is in rowech...
 7.109: In Exercises 109112, find the determinant of the matrix. 8254
 7.110: In Exercises 109112, find the determinant of the matrix. 97114 8
 7.111: In Exercises 109112, find the determinant of the matrix. 5010305
 7.112: In Exercises 109112, find the determinant of the matrix. 14122415 50
 7.113: In Exercises 113116, find all the (a) minors and (b) cofactors of t...
 7.114: In Exercises 113116, find all the (a) minors and (b) cofactors of t...
 7.115: In Exercises 113116, find all the (a) minors and (b) cofactors of t...
 7.116: In Exercises 113116, find all the (a) minors and (b) cofactors of t...
 7.117: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.118: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.119: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.120: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.121: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.122: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.123: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.124: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.125: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.126: In Exercises 117126, find the determinant of the matrix. Expand by ...
 7.127: In Exercises 127130, use Cramers Rule to solve (if possible) the sy...
 7.128: In Exercises 127130, use Cramers Rule to solve (if possible) the sy...
 7.129: In Exercises 127130, use Cramers Rule to solve (if possible) the sy...
 7.130: In Exercises 127130, use Cramers Rule to solve (if possible) the sy...
 7.131: In Exercises 131134, use a determinant to find the area of the tria...
 7.132: In Exercises 131134, use a determinant to find the area of the tria...
 7.133: In Exercises 131134, use a determinant to find the area of the tria...
 7.134: In Exercises 131134, use a determinant to find the area of the tria...
 7.135: In Exercises 135 and 136, use a determinant to determine whether th...
 7.136: In Exercises 135 and 136, use a determinant to determine whether th...
 7.137: In Exercises 137140, use a determinant to find an equation of the l...
 7.138: In Exercises 137140, use a determinant to find an equation of the l...
 7.139: In Exercises 137140, use a determinant to find an equation of the l...
 7.140: In Exercises 137140, use a determinant to find an equation of the l...
 7.141: In Exercises 141 and 142, (a) write the uncoded row matrices for th...
 7.142: In Exercises 141 and 142, (a) write the uncoded row matrices for th...
 7.143: In Exercises 143 and 144, decode the cryptogram by using the invers...
 7.144: In Exercises 143 and 144, decode the cryptogram by using the invers...
 7.145: In Exercises 145 and 146, determine whether the statement is true o...
 7.146: In Exercises 145 and 146, determine whether the statement is true o...
 7.147: Using a Graphing Utility Use the matrix capabilities of a graphing ...
 7.148: Invertible Matrices Under what conditions does a matrix have an inv...
 7.149: Writing What is meant by the cofactor of an entry of a matrix? How ...
 7.150: Think About It Three people were asked to solve a system of equatio...
 7.151: Think About It Describe the rowechelon form of an augmented matrix...
 7.152: Solving an Equation Solve the equation for 2 3 5 8 0 .
Solutions for Chapter 7: Matrices and Determinants
Full solutions for College Algebra  9th Edition
ISBN: 9781133963028
Solutions for Chapter 7: Matrices and Determinants
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9781133963028. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 603 problems in chapter 7: Matrices and Determinants have been answered, more than 39669 students have viewed full stepbystep solutions from this chapter. Chapter 7: Matrices and Determinants includes 603 full stepbystep solutions.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).