 4.7.1: Determine whether each pair of matrices are inverses of each other.
 4.7.2: Determine whether each pair of matrices are inverses of each other.
 4.7.3: Determine whether each pair of matrices are inverses of each other.
 4.7.4: Determine whether each pair of matrices are inverses of each other.
 4.7.5: Find the inverse of each matrix, if it exists.
 4.7.6: Find the inverse of each matrix, if it exists.
 4.7.7: Find the inverse of each matrix, if it exists.
 4.7.8: Code a message using your own coding matrix. Give your message and ...
 4.7.9: Determine whether each pair of matrices are inverses of each other.
 4.7.10: Determine whether each pair of matrices are inverses of each other.
 4.7.11: Determine whether each pair of matrices are inverses of each other.
 4.7.12: Determine whether each pair of matrices are inverses of each other.
 4.7.13: Find the inverse of each matrix, if it exists.
 4.7.14: Find the inverse of each matrix, if it exists.
 4.7.15: Find the inverse of each matrix, if it exists.
 4.7.16: Find the inverse of each matrix, if it exists.
 4.7.17: Find the inverse of each matrix, if it exists.
 4.7.18: Find the inverse of each matrix, if it exists.
 4.7.19: Find the inverse of each matrix, if it exists.
 4.7.20: Find the inverse of each matrix, if it exists.
 4.7.21: Find the inverse of each matrix, if it exists.
 4.7.22: For Exercises 2224, use the alphabet table at the right. Your frien...
 4.7.23: For Exercises 2224, use the alphabet table at the right. Your frien...
 4.7.24: For Exercises 2224, use the alphabet table at the right. Your frien...
 4.7.25: Use the Internet or other reference to find examples of codes used ...
 4.7.26: Only square matrices have multiplicative identities.
 4.7.27: Only square matrices have multiplicative inverses.
 4.7.28: Some square matrices do not have multiplicative inverses
 4.7.29: Some square matrices do not have multiplicative identities.
 4.7.30: Determine whether each pair of matrices are inverses of each other.
 4.7.31: Determine whether each pair of matrices are inverses of each other.
 4.7.32: Find the inverse of each matrix, if it exists
 4.7.33: Find the inverse of each matrix, if it exists
 4.7.34: Find the inverse of each matrix, if it exists
 4.7.35: Compare the matrix used to reflect a figure over the xaxis to the ...
 4.7.36: The matrix used to rotate a figure 270 counterclockwise about the o...
 4.7.37: Write the vertex matrix A for the rectangle.
 4.7.38: Use matrix multiplication to find BA if B = 2 0 0 2
 4.7.39: Graph the vertices of the transformed rectangle. Describe the trans...
 4.7.40: Make a conjecture about what transformation B 1 describes on a coo...
 4.7.41: Find B 1 and multiply it by BA. Make a drawing to verify your conj...
 4.7.42: The key on a TI83/84 Plus graphing calculator is used to find the ...
 4.7.43: The key on a TI83/84 Plus graphing calculator is used to find the ...
 4.7.44: The key on a TI83/84 Plus graphing calculator is used to find the ...
 4.7.45: Explain how to find the inverse of a 2 2 matrix
 4.7.46: Create a square matrix that does not have an inverse. Explain how y...
 4.7.47: For which values of a, b, c, and d will A = a c b d = A 1 ?
 4.7.48: Use the information about cryptography on page 208 to explain how i...
 4.7.49: The message MEET_ME_ TOMORROW is converted into numbers (0 = space,...
 4.7.50: The message MEET_ME_ TOMORROW is converted into numbers (0 = space,...
 4.7.51: Use Cramers Rule to solve each system of equations.
 4.7.52: Use Cramers Rule to solve each system of equations.
 4.7.53: Use Cramers Rule to solve each system of equations.
 4.7.54: Evaluate each determinant.
 4.7.55: Evaluate each determinant.
 4.7.56: Evaluate each determinant.
 4.7.57: Find each product, if possible
 4.7.58: Find each product, if possible
 4.7.59: Find each product, if possible
 4.7.60: Solve each system of equations
 4.7.61: Solve each system of equations
 4.7.62: Solve each system of equations
 4.7.63: Find the slope of the line that passes through each pair of points. (
 4.7.64: Find the slope of the line that passes through each pair of points. (
 4.7.65: Find the slope of the line that passes through each pair of points. (
 4.7.66: Find the slope of the line that passes through each pair of points. (
 4.7.67: Find the slope of the line that passes through each pair of points. (
 4.7.68: Find the slope of the line that passes through each pair of points. (
 4.7.69: The bottom of the Mariana Trench in the Pacific Ocean is 6.8 miles ...
 4.7.70: Solve each equation
 4.7.71: Solve each equation
 4.7.72: Solve each equation
 4.7.73: Solve each equation
 4.7.74: Solve each equation
 4.7.75: Solve each equation
Solutions for Chapter 4.7: Identity and Inverse Matrices
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 4.7: Identity and Inverse Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. Since 75 problems in chapter 4.7: Identity and Inverse Matrices have been answered, more than 44681 students have viewed full stepbystep solutions from this chapter. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Chapter 4.7: Identity and Inverse Matrices includes 75 full stepbystep solutions.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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 .

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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 ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.