 8.1: In Exercises 110, solve the system by the method of substitution. 1...
 8.2: In Exercises 110, solve the system by the method of substitution. 2...
 8.3: In Exercises 110, solve the system by the method of substitution. 3...
 8.4: In Exercises 110, solve the system by the method of substitution. 4...
 8.5: In Exercises 110, solve the system by the method of substitution. 5...
 8.6: In Exercises 110, solve the system by the method of substitution. 6...
 8.7: In Exercises 110, solve the system by the method of substitution. 7...
 8.8: In Exercises 110, solve the system by the method of substitution. 8...
 8.9: In Exercises 110, solve the system by the method of substitution. 9...
 8.10: In Exercises 110, solve the system by the method of substitution. 1...
 8.11: In Exercises 1116, use a graphing utility to approximate all points...
 8.12: In Exercises 1116, use a graphing utility to approximate all points...
 8.13: In Exercises 1116, use a graphing utility to approximate all points...
 8.14: In Exercises 1116, use a graphing utility to approximate all points...
 8.15: In Exercises 1116, use a graphing utility to approximate all points...
 8.16: In Exercises 1116, use a graphing utility to approximate all points...
 8.17: You invest $5000 in a greenhouse. The planter, potting soil, and se...
 8.18: You are offered two sales jobs. One company offers an annual salary...
 8.19: The perimeter of a rectangle is 480 meters and its length is 1.5 ti...
 8.20: The perimeter of a rectangle is 68 feet and its width is 8 9 times ...
 8.21: In Exercises 2130, solve the system by the method of elimination. 2...
 8.22: In Exercises 2130, solve the system by the method of elimination. 2...
 8.23: In Exercises 2130, solve the system by the method of elimination. 2...
 8.24: In Exercises 2130, solve the system by the method of elimination. 2...
 8.25: In Exercises 2130, solve the system by the method of elimination. 2...
 8.26: In Exercises 2130, solve the system by the method of elimination. 2...
 8.27: In Exercises 2130, solve the system by the method of elimination. 2...
 8.28: In Exercises 2130, solve the system by the method of elimination.28...
 8.29: In Exercises 2130, solve the system by the method of elimination. 2...
 8.30: In Exercises 2130, solve the system by the method of elimination. 3...
 8.31: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.32: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.33: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.34: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.35: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.36: In Exercises 3136, use a graphing utility to graph the lines in the...
 8.37: In Exercises 37 and 38, find the point of equilibrium of the demand...
 8.38: In Exercises 37 and 38, find the point of equilibrium of the demand...
 8.39: Two planes leave Pittsburgh and Philadelphia at the same time, each...
 8.40: A total of $46,000 is invested in two corporate bonds that pay 6.75...
 8.41: In Exercises 41 and 42, use backsubstitution to solve the system o...
 8.42: In Exercises 41 and 42, use backsubstitution to solve the system o...
 8.43: In Exercises 4350, solve the system of linear equations and check a...
 8.44: In Exercises 4350, solve the system of linear equations and check a...
 8.45: In Exercises 4350, solve the system of linear equations and check a...
 8.46: In Exercises 4350, solve the system of linear equations and check a...
 8.47: In Exercises 4350, solve the system of linear equations and check a...
 8.48: In Exercises 4350, solve the system of linear equations and check a...
 8.49: In Exercises 4350, solve the system of linear equations and check a...
 8.50: In Exercises 4350, solve the system of linear equations and check a...
 8.51: In Exercises 51 and 52, sketch the plane represented by the linear ...
 8.52: In Exercises 51 and 52, sketch the plane represented by the linear ...
 8.53: In Exercises 5358, write the partial fraction decomposition for the...
 8.54: In Exercises 5358, write the partial fraction decomposition for the...
 8.55: In Exercises 5358, write the partial fraction decomposition for the...
 8.56: In Exercises 5358, write the partial fraction decomposition for the...
 8.57: In Exercises 5358, write the partial fraction decomposition for the...
 8.58: In Exercises 5358, write the partial fraction decomposition for the...
 8.59: In Exercises 59 and 60, find the equation of the parabola y = ax2 +...
 8.60: In Exercises 59 and 60, find the equation of the parabola y = ax2 +...
 8.61: Pebble Beach Golf Links in Pebble Beach, California, is an 18hole ...
 8.62: An inheritance of $40,000 is divided among three investments yieldi...
 8.63: In Exercises 6366, determine the dimension of the matrix. 63. [ 3 1...
 8.64: In Exercises 6366, determine the dimension of the matrix. 63. [ 3 1...
 8.65: In Exercises 6366, determine the dimension of the matrix. 63. [ 3 1...
 8.66: In Exercises 6366, determine the dimension of the matrix. 63. [ 3 1...
 8.67: In Exercises 6770, write the augmented matrix for the system of lin...
 8.68: In Exercises 6770, write the augmented matrix for the system of lin...
 8.69: In Exercises 6770, write the augmented matrix for the system of lin...
 8.70: In Exercises 6770, write the augmented matrix for the system of lin...
 8.71: In Exercises 71 and 72, write the system of linear equations repres...
 8.72: In Exercises 71 and 72, write the system of linear equations repres...
 8.73: In Exercises 7376, write the matrix in rowechelon form. Remember t...
 8.74: In Exercises 7376, write the matrix in rowechelon form. Remember t...
 8.75: In Exercises 7376, write the matrix in rowechelon form. Remember t...
 8.76: In Exercises 7376, write the matrix in rowechelon form. Remember t...
 8.77: In Exercises 7780, use the matrix capabilities of a graphing utilit...
 8.78: In Exercises 7780, use the matrix capabilities of a graphing utilit...
 8.79: In Exercises 7780, use the matrix capabilities of a graphing utilit...
 8.80: In Exercises 7780, use the matrix capabilities of a graphing utilit...
 8.81: In Exercises 8188, use matrices to solve the system of equations, i...
 8.82: In Exercises 8188, use matrices to solve the system of equations, i...
 8.83: In Exercises 8188, use matrices to solve the system of equations, i...
 8.84: In Exercises 8188, use matrices to solve the system of equations, i...
 8.85: In Exercises 8188, use matrices to solve the system of equations, i...
 8.86: In Exercises 8188, use matrices to solve the system of equations, i...
 8.87: In Exercises 8188, use matrices to solve the system of equations, i...
 8.88: In Exercises 8188, use matrices to solve the system of equations, i...
 8.89: In Exercises 8996, use matrices to solve the system of equations, i...
 8.90: In Exercises 8996, use matrices to solve the system of equations, i...
 8.91: In Exercises 8996, use matrices to solve the system of equations, i...
 8.92: In Exercises 8996, use matrices to solve the system of equations, i...
 8.93: In Exercises 8996, use matrices to solve the system of equations, i...
 8.94: In Exercises 8996, use matrices to solve the system of equations, i...
 8.95: In Exercises 8996, use matrices to solve the system of equations, i...
 8.96: In Exercises 8996, use matrices to solve the system of equations, i...
 8.97: In Exercises 97100, use the matrix capabilities of a graphing utili...
 8.98: In Exercises 97100, use the matrix capabilities of a graphing utili...
 8.99: In Exercises 97100, use the matrix capabilities of a graphing utili...
 8.100: In Exercises 97100, use the matrix capabilities of a graphing utili...
 8.101: In Exercises 101104, find x and y. 101. [ 1 y x 9] = [ 1 7 12 9] 10...
 8.102: In Exercises 101104, find x and y. 101. [ 1 y x 9] = [ 1 7 12 9] 10...
 8.103: In Exercises 101104, find x and y. 101. [ 1 y x 9] = [ 1 7 12 9] 10...
 8.104: In Exercises 101104, find x and y. 101. [ 1 y x 9] = [ 1 7 12 9] 10...
 8.105: In Exercises 105108, find, if possible, (a) A + B, (b) A B, (c) 2A,...
 8.106: In Exercises 105108, find, if possible, (a) A + B, (b) A B, (c) 2A,...
 8.107: In Exercises 105108, find, if possible, (a) A + B, (b) A B, (c) 2A,...
 8.108: In Exercises 105108, find, if possible, (a) A + B, (b) A B, (c) 2A,...
 8.109: In Exercises 109112, evaluate the expression. 109. [ 2 0 1 5 0 4] 3...
 8.110: In Exercises 109112, evaluate the expression. 109. [ 2 0 1 5 0 4] 3...
 8.111: In Exercises 109112, evaluate the expression. 109. [ 2 0 1 5 0 4] 3...
 8.112: In Exercises 109112, evaluate the expression. 109. [ 2 0 1 5 0 4] 3...
 8.113: In Exercises 113 and 114, use the matrix capabilities of a graphing...
 8.114: In Exercises 113 and 114, use the matrix capabilities of a graphing...
 8.115: In Exercises 115118, solve for X when A = [ 4 1 3 0 5 2 ] and B = [...
 8.116: In Exercises 115118, solve for X when A = [ 4 1 3 0 5 2 ] and B = [...
 8.117: In Exercises 115118, solve for X when A = [ 4 1 3 0 5 2 ] and B = [...
 8.118: In Exercises 115118, solve for X when A = [ 4 1 3 0 5 2 ] and B = [...
 8.119: In Exercises 119122, find AB, if possible. 119. A = [ 1 5 6 2 4 0 ]...
 8.120: In Exercises 119122, find AB, if possible. 119. A = [ 1 5 6 2 4 0 ]...
 8.121: In Exercises 119122, find AB, if possible. 119. A = [ 1 5 6 2 4 0 ]...
 8.122: In Exercises 119122, find AB, if possible. 119. A = [ 1 5 6 2 4 0 ]...
 8.123: In Exercises 119122, find AB, if possible. 119. A = [ 1 5 6 2 4 0 ]...
 8.124: In Exercises 123126, use the matrix capabilities of a graphing util...
 8.125: In Exercises 123126, use the matrix capabilities of a graphing util...
 8.126: In Exercises 123126, use the matrix capabilities of a graphing util...
 8.127: A tire corporation has three factories, each of which manufactures ...
 8.128: An electronics manufacturing company produces three different model...
 8.129: In Exercises 129 and 130, show that B is the inverse of A. 129. A =...
 8.130: In Exercises 129 and 130, show that B is the inverse of A. 129. A =...
 8.131: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.132: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.133: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.134: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.135: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.136: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.137: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.138: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.139: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.140: In Exercises 131134, find the inverse of the matrix (if it exists)....
 8.141: In Exercises 141146, use the formula on page 676 to find the invers...
 8.142: In Exercises 141146, use the formula on page 676 to find the invers...
 8.143: In Exercises 141146, use the formula on page 676 to find the invers...
 8.144: In Exercises 141146, use the formula on page 676 to find the invers...
 8.145: In Exercises 141146, use the formula on page 676 to find the invers...
 8.146: In Exercises 141146, use the formula on page 676 to find the invers...
 8.147: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.148: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.149: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.150: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.151: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.152: In Exercises 147152, use an inverse matrix to solve (if possible) t...
 8.153: In Exercises 153156, use the matrix capabilities of a graphing util...
 8.154: In Exercises 153156, use the matrix capabilities of a graphing util...
 8.155: In Exercises 153156, use the matrix capabilities of a graphing util...
 8.156: In Exercises 153156, use the matrix capabilities of a graphing util...
 8.157: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.158: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.159: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.160: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.161: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.162: In Exercises 157162, find the determinant of the matrix. 157. [23] ...
 8.163: In Exercises 163166, find all (a) minors and (b) cofactors of the m...
 8.164: In Exercises 163166, find all (a) minors and (b) cofactors of the m...
 8.165: In Exercises 163166, find all (a) minors and (b) cofactors of the m...
 8.166: In Exercises 163166, find all (a) minors and (b) cofactors of the m...
 8.167: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.168: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.169: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.170: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.171: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.172: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.173: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.174: In Exercises 167174, find the determinant of the matrix. Expand by ...
 8.175: In Exercises 175180, use a determinant to find the area of the figu...
 8.176: In Exercises 175180, use a determinant to find the area of the figu...
 8.177: In Exercises 175180, use a determinant to find the area of the figu...
 8.178: In Exercises 175180, use a determinant to find the area of the figu...
 8.179: In Exercises 175180, use a determinant to find the area of the figu...
 8.180: In Exercises 175180, use a determinant to find the area of the figu...
 8.181: In Exercises 181184, use a determinant to determine whether the poi...
 8.182: In Exercises 181184, use a determinant to determine whether the poi...
 8.183: In Exercises 181184, use a determinant to determine whether the poi...
 8.184: In Exercises 181184, use a determinant to determine whether the poi...
 8.185: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.186: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.187: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.188: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.189: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.190: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.191: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.192: In Exercises 185192, use Cramers Rule to solve (if possible) the sy...
 8.193: In Exercises 193 and 194, solve the system of equations using (a) G...
 8.194: In Exercises 193 and 194, solve the system of equations using (a) G...
 8.195: In Exercises 195 and 196, (a) write the uncoded 1 3 row matrices fo...
 8.196: In Exercises 195 and 196, (a) write the uncoded 1 3 row matrices fo...
 8.197: In Exercises 197199, use A1 to decode the cryptogram. 197. A = [ 1 ...
 8.198: In Exercises 197199, use A1 to decode the cryptogram. 197. A = [ 1 ...
 8.199: In Exercises 197199, use A1 to decode the cryptogram. 197. A = [ 1 ...
 8.200: The populations (in millions) of Florida for the years 2010 through...
 8.201: In Exercises 201 and 202, determine whether the statement is true o...
 8.202: In Exercises 201 and 202, determine whether the statement is true o...
 8.203: What is the relationship between the three elementary row operation...
 8.204: Under what conditions does a matrix have an inverse?
Solutions for Chapter 8: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8: Linear Systems and Matrices
Get Full SolutionsChapter 8: Linear Systems and Matrices includes 204 full stepbystep solutions. Since 204 problems in chapter 8: Linear Systems and Matrices have been answered, more than 59308 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry: Real Mathematics, Real People was written by and is associated to the ISBN: 9781305071735. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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].

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.