 47.1: A = 2 1 1 3 , B = _1 2 0 0  _1 3
 47.2: X = 3 5 1 2 , Y = 2 5 1 3
 47.3: C = 1 0 1 1 , D = 1 0 1 1
 47.4: F = 3 4 1 2 , G = 1 3 2 4
 47.5: 8 3 5 2
 47.6: 4 1 8 2
 47.7: 5 7 1 4
 47.8: CRYPTOGRAPHY Code a message using your own coding matrix. Give your...
 47.9: P = 0 1 1 1 , Q = 1 1 1 0
 47.10: R = 2 3 2 4 , S = 2  _3 2 1 1
 47.11: A = 6 5 2 2 , B = 1  _5 2 1 3
 47.12: X = _1 3 _2 3  _2 3  _1 3 , Y = 1 2 2 1
 47.13: 5 0 0 1
 47.14: 1 2 2 1
 47.15: 6 8 3 4
 47.16: 3 6 2 4
 47.17: 3 4 1 1
 47.18: 2 7 6
 47.19: 4 2 3 7
 47.20: 2 5 0 6
 47.21: 4 6 6 9
 47.22: 50  36  51  29  18  18  26  13  33  26  44  22  48  33...
 47.23: 59  33  8  8  39  21  7  7  56  37  25  16  4  2
 47.24: 59  34  49  31  40  20  16  14  21  15  25  25  36  24...
 47.25: RESEARCH Use the Internet or other reference to find examples of co...
 47.26: Only square matrices have multiplicative identities.
 47.27: Only square matrices have multiplicative inverses.
 47.28: Some square matrices do not have multiplicative inverses.
 47.29: Some square matrices do not have multiplicative identities.
 47.30: C = 1 1 5 2 , D = _2 7 _1 7 _5 7  _1 7
 47.31: J = 1 2 1 2 3 1 3 1 2 , K =  _5 4 _3 4 _1 4 _1 4 _1 4  _1 4 _7 4 ...
 47.32: 2 6 5 1
 47.33: _1 2 _1 6  _3 4 _1 4
 47.34: _3 10 _1 5 _5 8 _3 4
 47.35: GEOMETRY Compare the matrix used to reflect a figure over the xaxi...
 47.36: GEOMETRY The matrix used to rotate a figure 270 counterclockwise ab...
 47.37: Write the vertex matrix A for the rectangle.
 47.38: Use matrix multiplication to find BA if B = 2 0 0 2 .
 47.39: Graph the vertices of the transformed rectangle. Describe the trans...
 47.40: Make a conjecture about what transformation B 1 describes on a coo...
 47.41: Find B 1 and multiply it by BA. Make a drawing to verify your conj...
 47.42: 11 6 9 5
 47.43: 12 15 4 5
 47.44: 3 2 3 1 0 5 2 4 2
 47.45: REASONING Explain how to find the inverse of a 2 2 matrix.
 47.46: OPEN ENDED Create a square matrix that does not have an inverse. Ex...
 47.47: CHALLENGE For which values of a, b, c, and d will A = a c b d = A 1 ?
 47.48: Writing in Math Use the information about cryptography on page 208 ...
 47.49: ACT/SAT The message MEET_ME_ TOMORROW is converted into numbers (0 ...
 47.50: REVIEW Line q is shown below. Which equation best represents a line...
 47.51: 3x + 2y = 2 52. 2x + 5y = 35 53. 4x 3z = 23 x  3y = 14
 47.52: 2x + 5y = 35 53. 4x 3z = 23 x  3y = 14 7x  4y = 28
 47.53: 4x 3z = 23 x  3y = 14 7x  4y = 28 2x 5y + z = 9 y z = 3
 47.54: 2 4 3 8 5 6 6 2 1
 47.55: 9 5 3 2 2 1 3 1
 47.56: 5 1 5 7 2 7 3 9 3
 47.57: [5 2] 2 3
 47.58: 7 1 3 4 2 5 [3 5]
 47.59: [4 2 0] 3 1 5 2 0 6
 47.60: 3x + 5y = 2 61. 6x + 2y = 22 62. 3x  2y = 2 2x  y = 3
 47.61: 6x + 2y = 22 62. 3x  2y = 2 2x  y = 3 3x + 7y = 41
 47.62: 3x  2y = 2 2x  y = 3 3x + 7y = 41 4x + 7y = 65
 47.63: (2, 5), (6, 9)
 47.64: (1, 0), (2, 9)
 47.65: (5, 4), (3, 6)
 47.66: (2, 2), (5, 1)
 47.67: (0, 3), (2, 2)
 47.68: (8, 9), (0, 6)
 47.69: OCEANOGRAPHY The bottom of the Mariana Trench in the Pacific Ocean ...
 47.70: 3k + 8 = 5
 47.71: 12 = 5h + 2
 47.72: 7z 4 = 5z + 8
 47.73: x 2 + 5 = 7
 47.74: 3 + n 6 = 4
 47.75: 6 = _s  8 7
Solutions for Chapter 47: Identity and Inverse Matrices
Full solutions for Algebra 2, Student Edition (MERRILL ALGEBRA 2)  1st Edition
ISBN: 9780078738302
Solutions for Chapter 47: Identity and Inverse Matrices
Get Full SolutionsChapter 47: Identity and Inverse Matrices includes 75 full stepbystep solutions. Since 75 problems in chapter 47: Identity and Inverse Matrices have been answered, more than 56532 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 2, Student Edition (MERRILL ALGEBRA 2), edition: 1. Algebra 2, Student Edition (MERRILL ALGEBRA 2) was written by and is associated to the ISBN: 9780078738302.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite 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.

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.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

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.

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.

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.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).