 8.5.1: In Exercises 14, fill in the blank(s). 1. Two matrices are _______ ...
 8.5.2: In Exercises 14, fill in the blank(s). 2. When working with matrice...
 8.5.3: In Exercises 14, fill in the blank(s). 3. A matrix consisting entir...
 8.5.4: In Exercises 14, fill in the blank(s). 4. The n n matrix consisting...
 8.5.5: In Exercises 5 and 6, match the matrix property with the correct fo...
 8.5.6: In Exercises 5 and 6, match the matrix property with the correct fo...
 8.5.7: In general, when multiplying matrices A and B, does AB = BA?
 8.5.8: What is the dimension of AB when A is a 2 3 matrix and B is a 3 4 m...
 8.5.9: In Exercises 912, find x and y or x, y, and z. 9. [ x 9 7 y] = [ 5 ...
 8.5.10: In Exercises 912, find x and y or x, y, and z. 10. [ 5 y x 8] = [ 5...
 8.5.11: In Exercises 912, find x and y or x, y, and z. 11. [ 4 13 2 5 15 2z...
 8.5.12: In Exercises 912, find x and y or x, y, and z. 12. [ x + 4 1 7 8 22...
 8.5.13: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.14: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.15: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.16: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.17: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.18: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.19: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.20: In Exercises 1320, find, if possible, (a) A + B, (b) A B, (c) 3A, a...
 8.5.21: In Exercises 2124, evaluate the expression. 21. [ 5 3 0 6] + [ 7 2 ...
 8.5.22: In Exercises 2124, evaluate the expression. 22. [ 6 1 7 9 0 1 ] + [...
 8.5.23: In Exercises 2124, evaluate the expression. 23. 1 3([ 4 0 0 2 1 12]...
 8.5.24: In Exercises 2124, evaluate the expression. 24. 1 2([3 2 4 0] [10 6...
 8.5.25: In Exercises 2528, use the matrix capabilities of a graphing utilit...
 8.5.26: In Exercises 2528, use the matrix capabilities of a graphing utilit...
 8.5.27: In Exercises 2528, use the matrix capabilities of a graphing utilit...
 8.5.28: In Exercises 2528, use the matrix capabilities of a graphing utilit...
 8.5.29: In Exercises 2932, solve for X when A = [ 2 1 3 1 0 4 ] and B = [ 0...
 8.5.30: In Exercises 2932, solve for X when A = [ 2 1 3 1 0 4 ] and B = [ 0...
 8.5.31: In Exercises 2932, solve for X when A = [ 2 1 3 1 0 4 ] and B = [ 0...
 8.5.32: In Exercises 2932, solve for X when A = [ 2 1 3 1 0 4 ] and B = [ 0...
 8.5.33: In Exercises 3340, find AB, if possible. 33. A = [ 3 4 2 1 5 6 ] , ...
 8.5.34: In Exercises 3340, find AB, if possible. 34. A = [ 1 4 0 6 5 3 ] , ...
 8.5.35: In Exercises 3340, find AB, if possible. 35. A = [ 2 3 1 1 4 6 ] , ...
 8.5.36: In Exercises 3340, find AB, if possible. 36. A = [ 1 6 0 13 3 8 2 1...
 8.5.37: In Exercises 3340, find AB, if possible. 37. A = [ 6 0 0 0 4 0 0 0 ...
 8.5.38: In Exercises 3340, find AB, if possible. 38. A = [ 5 0 0 0 8 0 0 0 ...
 8.5.39: In Exercises 3340, find AB, if possible. 39. A = [ 5 3 4 ] , B = [2...
 8.5.40: In Exercises 3340, find AB, if possible. 40. A = [ 5 6], B = [3 1 5 9]
 8.5.41: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.42: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.43: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.44: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.45: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.46: In Exercises 4146, find, if possible, (a) AB, (b) BA, and (c) A2. (...
 8.5.47: In Exercises 4750, use the matrix capabilities of a graphing utilit...
 8.5.48: In Exercises 4750, use the matrix capabilities of a graphing utilit...
 8.5.49: In Exercises 4750, use the matrix capabilities of a graphing utilit...
 8.5.50: In Exercises 4750, use the matrix capabilities of a graphing utilit...
 8.5.51: In Exercises 5154, use the matrix capabilities of a graphing utilit...
 8.5.52: In Exercises 5154, use the matrix capabilities of a graphing utilit...
 8.5.53: In Exercises 5154, use the matrix capabilities of a graphing utilit...
 8.5.54: In Exercises 5154, use the matrix capabilities of a graphing utilit...
 8.5.55: In Exercises 5558, use matrix multiplication to determine whether e...
 8.5.56: In Exercises 5558, use matrix multiplication to determine whether e...
 8.5.57: In Exercises 5558, use matrix multiplication to determine whether e...
 8.5.58: In Exercises 5558, use matrix multiplication to determine whether e...
 8.5.59: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.60: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.61: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.62: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.63: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.64: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.65: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.66: In Exercises 5966, (a) write the system of equations as a matrix eq...
 8.5.67: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.68: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.69: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.70: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.71: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.72: In Exercises 6772, use a graphing utility to perform the operations...
 8.5.73: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.74: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.75: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.76: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.77: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.78: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.79: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.80: In Exercises 7380, perform the operations (a) using a graphing util...
 8.5.81: In Exercises 81 and 82, use a graphing utility to perform the indic...
 8.5.82: In Exercises 81 and 82, use a graphing utility to perform the indic...
 8.5.83: A corporation that makes sunglasses has four factories, each of whi...
 8.5.84: A corporation has four factories, each of which manufactures sport ...
 8.5.85: A corporation has three factories, each of which manufactures acous...
 8.5.86: A vacation service has identified four resort hotels with a special...
 8.5.87: A fruit grower raises two crops, apples and peaches. Each of these ...
 8.5.88: The numbers of calories burned per hour by individuals of different...
 8.5.89: A company manufactures boats. Its laborhour and wage requirements ...
 8.5.90: A company sells five models of computers through three retail outle...
 8.5.91: Use a graphing utility to find P3, P4, P5, P6, P7, and P8 for the m...
 8.5.92: Use a graphing utility to find P3, P4, P5, P6, P7, and P8 for the m...
 8.5.93: In Exercises 93 and 94, determine whether the statement is true or ...
 8.5.94: In Exercises 93 and 94, determine whether the statement is true or ...
 8.5.95: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.96: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.97: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.98: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.99: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.100: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.101: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.102: In Exercises 95102, let matrices A, B, C, and D be of dimensions 2 ...
 8.5.103: In Exercises 103106, use the matrices A = [ 2 1 1 3] and B = [ 1 0 ...
 8.5.104: In Exercises 103106, use the matrices A = [ 2 1 1 3] and B = [ 1 0 ...
 8.5.105: In Exercises 103106, use the matrices A = [ 2 1 1 3] and B = [ 1 0 ...
 8.5.106: In Exercises 103106, use the matrices A = [ 2 1 1 3] and B = [ 1 0 ...
 8.5.107: If a, b, and c are real numbers such that c 0 and ac = bc, then a =...
 8.5.108: If a and b are real numbers such that ab = 0, then a = 0 or b = 0. ...
 8.5.109: Let i = 1 and let A = [ i 0 0 i ] and B = [ 0 i i 0]. (a) Find A2, ...
 8.5.110: Let A and B be unequal diagonal matrices of the same dimension. (A ...
 8.5.111: Let matrices A and B be of dimensions 3 2 and 2 2, respectively. An...
 8.5.112: An electronics manufacturer produces two models of LCD televisions,...
 8.5.113: In Exercises 113 and 114, condense the expression to the logarithm ...
 8.5.114: In Exercises 113 and 114, condense the expression to the logarithm ...
Solutions for Chapter 8.5: Linear Systems and Matrices
Full solutions for Algebra and Trigonometry: Real Mathematics, Real People  7th Edition
ISBN: 9781305071735
Solutions for Chapter 8.5: Linear Systems and Matrices
Get Full SolutionsChapter 8.5: Linear Systems and Matrices includes 114 full stepbystep solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry: Real Mathematics, Real People, edition: 7. Since 114 problems in chapter 8.5: Linear Systems and Matrices have been answered, more than 61405 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 expansive textbook survival guide covers the following chapters and their solutions.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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

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.

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

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

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.