 7.4.7.4.1: A rectangular array of real numbers that can be used to solve a sys...
 7.4.7.4.2: A matrix is _______ if the number of rows equals the number of colu...
 7.4.7.4.3: A matrix with only one row is called a _______ and a matrix with on...
 7.4.7.4.4: The matrix derived from a system of linear equations is called the ...
 7.4.7.4.5: The matrix derived from the coefficients of a system of linear equa...
 7.4.7.4.6: Two matrices are called _______ if one of the matrices can be obtai...
 7.4.7.4.7: A matrix in rowechelon form is in _______ if every column that has...
 7.4.7.4.8: can't copypaste the problem
 7.4.7.4.9: In Exercises 710, write the augmented matrix for the system of line...
 7.4.7.4.10: In Exercises 710, write the augmented matrix for the system of line...
 7.4.7.4.11: In Exercises 1114, write the system of linear equations represented...
 7.4.7.4.12: In Exercises 1114, write the system of linear equations represented...
 7.4.7.4.13: In Exercises 1114, write the system of linear equations represented...
 7.4.7.4.14: In Exercises 1114, write the system of linear equations represented...
 7.4.7.4.15: In Exercises 1518, fill in the blanks using elementary row operatio...
 7.4.7.4.16: In Exercises 1518, fill in the blanks using elementary row operatio...
 7.4.7.4.17: In Exercises 1518, fill in the blanks using elementary row operatio...
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 7.4.7.4.19: In Exercises 1922, identify the elementary row operation performed ...
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 7.4.7.4.21: In Exercises 1922, identify the elementary row operation performed ...
 7.4.7.4.22: In Exercises 1922, identify the elementary row operation performed ...
 7.4.7.4.23: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.24: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.25: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.26: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.27: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.28: In Exercises 2328, determine whether the matrix is in rowechelon f...
 7.4.7.4.29: Perform the sequence of row operations on the matrix. What did the ...
 7.4.7.4.30: Perform the sequence of row operations on the matrix. What did the ...
 7.4.7.4.31: Repeat steps (a) through (e) in Exercise 29 using a graphing utility.
 7.4.7.4.32: Repeat steps (a) through (f) in Exercise 30 using a graphing utility
 7.4.7.4.33: In Exercises 3336, write the matrix in rowechelon form. Remember t...
 7.4.7.4.34: In Exercises 3336, write the matrix in rowechelon form. Remember t...
 7.4.7.4.35: In Exercises 3336, write the matrix in rowechelon form. Remember t...
 7.4.7.4.36: In Exercises 3336, write the matrix in rowechelon form. Remember t...
 7.4.7.4.37: In Exercises 3740, use the matrix capabilities of a graphing utilit...
 7.4.7.4.38: In Exercises 3740, use the matrix capabilities of a graphing utilit...
 7.4.7.4.39: In Exercises 3740, use the matrix capabilities of a graphing utilit...
 7.4.7.4.40: In Exercises 3740, use the matrix capabilities of a graphing utilit...
 7.4.7.4.41: In Exercises 41 44, write the system of linear equations represente...
 7.4.7.4.42: In Exercises 41 44, write the system of linear equations represente...
 7.4.7.4.43: In Exercises 41 44, write the system of linear equations represente...
 7.4.7.4.44: In Exercises 41 44, write the system of linear equations represente...
 7.4.7.4.45: In Exercises 45 48, an augmented matrix that represents a system of...
 7.4.7.4.46: In Exercises 45 48, an augmented matrix that represents a system of...
 7.4.7.4.47: In Exercises 45 48, an augmented matrix that represents a system of...
 7.4.7.4.48: In Exercises 45 48, an augmented matrix that represents a system of...
 7.4.7.4.49: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.50: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.51: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.52: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.53: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.54: In Exercises 4954, use matrices to solve the system of equations, i...
 7.4.7.4.55: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.56: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.57: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.58: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.59: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.60: In Exercises 55 60, use matrices to solve the system of equations, ...
 7.4.7.4.61: In Exercises 61 64, use the matrix capabilities of a graphing utili...
 7.4.7.4.62: In Exercises 61 64, use the matrix capabilities of a graphing utili...
 7.4.7.4.63: In Exercises 61 64, use the matrix capabilities of a graphing utili...
 7.4.7.4.64: In Exercises 61 64, use the matrix capabilities of a graphing utili...
 7.4.7.4.65: In Exercises 65 68, determine whether the two systems of linear equ...
 7.4.7.4.66: In Exercises 65 68, determine whether the two systems of linear equ...
 7.4.7.4.67: In Exercises 65 68, determine whether the two systems of linear equ...
 7.4.7.4.68: In Exercises 65 68, determine whether the two systems of linear equ...
 7.4.7.4.69: In Exercises 6972, use a system of equations to find the equation o...
 7.4.7.4.70: In Exercises 6972, use a system of equations to find the equation o...
 7.4.7.4.71: In Exercises 6972, use a system of equations to find the equation o...
 7.4.7.4.72: In Exercises 6972, use a system of equations to find the equation o...
 7.4.7.4.73: In Exercises 73 and 74, use a system of equations to find the quadr...
 7.4.7.4.74: In Exercises 73 and 74, use a system of equations to find the quadr...
 7.4.7.4.75: In Exercises 75 and 76, use a system of equations to find the cubic...
 7.4.7.4.76: In Exercises 75 and 76, use a system of equations to find the cubic...
 7.4.7.4.77: Borrowing Money A small corporation borrowed $1,500,000 to expand i...
 7.4.7.4.78: Borrowing Money A small corporation borrowed $500,000 to build a ne...
 7.4.7.4.79: Electrical Network The currents in an electrical network are given ...
 7.4.7.4.80: Mathematical Modeling A videotape of the path of a ball thrown by a...
 7.4.7.4.81: Data Analysis The table shows the average retail prices (in dollars...
 7.4.7.4.82: Data Analysis The table shows the average annual salaries (in thous...
 7.4.7.4.83: Paper A wholesale paper company sells a 100pound package of comput...
 7.4.7.4.84: Tickets The theater department of a high school has collected the r...
 7.4.7.4.85: Network Analysis In Exercises 85 and 86, answer the questions about...
 7.4.7.4.86: Network Analysis In Exercises 85 and 86, answer the questions about...
 7.4.7.4.87: True or False? In Exercises 87 and 88, determine whether the statem...
 7.4.7.4.88: True or False? In Exercises 87 and 88, determine whether the statem...
 7.4.7.4.89: Think About It The augmented matrix represents a system of linear e...
 7.4.7.4.90: Think About It (a) Describe the rowechelon form of an augmented ma...
 7.4.7.4.91: rror Analysis One of your classmates has submitted the following st...
 7.4.7.4.92: Writing In your own words, describe the difference between a matrix...
 7.4.7.4.93: In Exercises 9396, sketch the graph of the function. Identify any a...
 7.4.7.4.94: In Exercises 9396, sketch the graph of the function. Identify any a...
 7.4.7.4.95: In Exercises 9396, sketch the graph of the function. Identify any a...
 7.4.7.4.96: In Exercises 9396, sketch the graph of the function. Identify any a...
Solutions for Chapter 7.4: Matrices and Systems of Equations
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 7.4: Matrices and Systems of Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 96 problems in chapter 7.4: Matrices and Systems of Equations have been answered, more than 48021 students have viewed full stepbystep solutions from this chapter. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 7.4: Matrices and Systems of Equations includes 96 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

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).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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