 12.4.1: A matrix that has the same number of rows as columns is called a(n)...
 12.4.2: True or False Matrix addition is commutative
 12.4.3: To find the product AB of two matrices A and B, the number of in ma...
 12.4.4: True or False Matrix multiplication is commutative.
 12.4.5: Suppose that A is a square n by n matrix that is nonsingular. The m...
 12.4.6: If a matrix A has no inverse, it is called
 12.4.7: True or False The identity matrix has properties similar to those o...
 12.4.8: If represents a matrix equation where A is a nonsingular matrix, th...
 12.4.9: In 924, use the following matrices to evaluate the given expression...
 12.4.10: In 924, use the following matrices to evaluate the given expression...
 12.4.11: In 924, use the following matrices to evaluate the given expression...
 12.4.12: In 924, use the following matrices to evaluate the given expression...
 12.4.13: In 924, use the following matrices to evaluate the given expression...
 12.4.14: In 924, use the following matrices to evaluate the given expression...
 12.4.15: In 924, use the following matrices to evaluate the given expression...
 12.4.16: In 924, use the following matrices to evaluate the given expression...
 12.4.17: In 924, use the following matrices to evaluate the given expression...
 12.4.18: In 924, use the following matrices to evaluate the given expression...
 12.4.19: In 924, use the following matrices to evaluate the given expression...
 12.4.20: In 924, use the following matrices to evaluate the given expression...
 12.4.21: In 924, use the following matrices to evaluate the given expression...
 12.4.22: In 924, use the following matrices to evaluate the given expression...
 12.4.23: In 924, use the following matrices to evaluate the given expression...
 12.4.24: In 924, use the following matrices to evaluate the given expression...
 12.4.25: In 2530, find the product
 12.4.26: In 2530, find the product
 12.4.27: In 2530, find the product
 12.4.28: In 2530, find the product
 12.4.29: In 2530, find the product
 12.4.30: In 2530, find the product
 12.4.31: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.32: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.33: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.34: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.35: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.36: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.37: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.38: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.39: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.40: In 3140, each matrix is nonsingular. Find the inverse of each matrix
 12.4.41: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.42: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.43: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.44: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.45: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.46: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.47: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.48: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.49: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.50: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.51: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.52: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.53: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.54: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.55: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.56: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.57: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.58: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.59: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.60: In 4160, use the inverses found in 3140 to solve each system of equ...
 12.4.61: In 6166,show that each matrix has no inverse.
 12.4.62: In 6166,show that each matrix has no inverse.
 12.4.63: In 6166,show that each matrix has no inverse.
 12.4.64: In 6166,show that each matrix has no inverse.
 12.4.65: In 6166,show that each matrix has no inverse.
 12.4.66: In 6166,show that each matrix has no inverse.
 12.4.67: In 6770, use a graphing utility to find the inverse, if it exists, ...
 12.4.68: In 6770, use a graphing utility to find the inverse, if it exists, ...
 12.4.69: In 6770, use a graphing utility to find the inverse, if it exists, ...
 12.4.70: In 6770, use a graphing utility to find the inverse, if it exists, ...
 12.4.71: In 7174, use the idea behind Example 15 with a graphing utility to ...
 12.4.72: In 7174, use the idea behind Example 15 with a graphing utility to ...
 12.4.73: In 7174, use the idea behind Example 15 with a graphing utility to ...
 12.4.74: In 7174, use the idea behind Example 15 with a graphing utility to ...
 12.4.75: In 7582, algebraically solve each system of equations using any met...
 12.4.76: In 7582, algebraically solve each system of equations using any met...
 12.4.77: In 7582, algebraically solve each system of equations using any met...
 12.4.78: In 7582, algebraically solve each system of equations using any met...
 12.4.79: In 7582, algebraically solve each system of equations using any met...
 12.4.80: In 7582, algebraically solve each system of equations using any met...
 12.4.81: In 7582, algebraically solve each system of equations using any met...
 12.4.82: In 7582, algebraically solve each system of equations using any met...
 12.4.83: College Tuition Nikki and Joe take classes at a community college, ...
 12.4.84: School Loan Interest Jamal and Stephanie each have school loans iss...
 12.4.85: Computing the Cost of Production The Acme Steel Company is a produc...
 12.4.86: Computing Profit Rizza Ford has two locations, one in the city and ...
 12.4.87: Cryptography One method of encryption is to use a matrix to encrypt...
 12.4.88: Economic Mobility The relative income of a child (low, medium, or h...
 12.4.89: Consider the 2 by 2 square matrix If show that A is nonsingular and...
 12.4.90: Create a situation different from any found in the text that can be...
 12.4.91: Explain why the number of columns in matrix A must equal the number...
 12.4.92: If a, b, and are real numbers with , then . Does this same property...
 12.4.93: What is the solution of the system of equations , if exists? Discus...
Solutions for Chapter 12.4: Algebra and Trigonometry 9th Edition
Full solutions for Algebra and Trigonometry  9th Edition
ISBN: 9780321716569
Solutions for Chapter 12.4
Get Full SolutionsSince 93 problems in chapter 12.4 have been answered, more than 61135 students have viewed full stepbystep solutions from this chapter. Chapter 12.4 includes 93 full stepbystep solutions. Algebra and Trigonometry was written by and is associated to the ISBN: 9780321716569. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 9.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

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

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.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.