 6.3.1: In Exercises 14, a. Give the order of each matrix. b. If A  [a;j]...
 6.3.2: In Exercises 14, a. Give the order of each matrix. b. If A  [a;j]...
 6.3.3: In Exercises 14, a. Give the order of each matrix. b. If A  [a;j]...
 6.3.4: In Exercises 14, a. Give the order of each matrix. b. If A  [a;j]...
 6.3.5: In Exercises 58, find tla/ues for the variables so drat the matric...
 6.3.6: In Exercises 58, find tla/ues for the variables so drat the matric...
 6.3.7: In Exercises 58, find tla/ues for the variables so drat the matric...
 6.3.8: In Exercises 58, find tla/ues for the variables so drat the matric...
 6.3.9: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.10: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.11: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.12: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.13: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.14: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.15: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.16: In Exercises 916, find the following matrices: a. A+ 8 b. A  8 c....
 6.3.17: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.18: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.19: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.20: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.21: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.22: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.23: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.24: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.25: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.26: In Exercises 1726, let An  9 7J and 8  r5 0 0 1 J . 0 3  4 ...
 6.3.27: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.28: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.29: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.30: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.31: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.32: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.33: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.34: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.35: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.36: In Exercises 2736, find (if possible) the followillg matrices: a. ...
 6.3.37: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.38: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.39: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.40: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.41: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.42: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.43: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.44: In Exercises 3744, perform the indicated matrix operations given t...
 6.3.45: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.46: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.47: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.48: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.49: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.50: In Exercises 4550, let 38. 5C  28 40. A(B + C) 42.8  A 44. A(CB)...
 6.3.51: The + sign in the figure is shown using 9 pixels in a 3 X 3 grid. T...
 6.3.52: The + sign in the figure is shown using 9 pixels in a 3 X 3 grid. T...
 6.3.53: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.54: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.55: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.56: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.57: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.58: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.59: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.60: The figure shows I he Jett.er L in a rectangular coordinate system....
 6.3.61: Completing the transition to adullhood is measured by one or more o...
 6.3.62: The table gives an estima te of basic caloric needs for diffe r(mt ...
 6.3.63: The final grade in a particular oourse is determined by grades on t...
 6.3.64: In a certain county, the proportion of voters in each age group reg...
 6.3.65: What is meant by the order of a matrix? Give an example with your e...
 6.3.66: What does a ij mean?
 6.3.67: What are equal matrices?
 6.3.68: How are matrices added?
 6.3.69: Describe how to subtract matrices.
 6.3.70: Describe matrices that cannot be added or subtracted.
 6.3.71: Describe how to perform scalar mulliplication. Provide an example w...
 6.3.72: Describe how to multiply matrices.
 6.3.73: Describe when the multiplication of two matrices is not defined.
 6.3.74: If two matrices can be multiplied, describe how to determine the or...
 6.3.75: Low resolution digital photographs use 262.11l pixels in a 512 X...
 6.3.76: Use the matrix feature of a graphing utility to verify each of your...
 6.3.77: In Exercises 7780, determine wlteJherroch staJemrnt makes J'ense o...
 6.3.78: In Exercises 7780, determine wlteJherroch staJemrnt makes J'ense o...
 6.3.79: In Exercises 7780, determine wlteJherroch staJemrnt makes J'ense o...
 6.3.80: In Exercises 7780, determine wlteJherroch staJemrnt makes J'ense o...
 6.3.81: Find two matrices A and 8 such that A8  8A.
 6.3.82: Consider a squnre matrix s uch that each element that is not on the...
 6.3.83: If A8  8A.then A and 8 are said to be antieommutativc. AreA ~ ~}...
 6.3.84: The interesting and uscfuJ applications of matrix theory are nearly...
 6.3.85: Exercises 8587 wilf help you prepare for the lllllle,;al covered i...
 6.3.86: Exercises 8587 wilf help you prepare for the lllllle,;al covered i...
 6.3.87: Exercises 8587 wilf help you prepare for the lllllle,;al covered i...
Solutions for Chapter 6.3: Matrix Operations and Their Applications
Full solutions for College Algebra  6th Edition
ISBN: 9780321782281
Solutions for Chapter 6.3: Matrix Operations and Their Applications
Get Full SolutionsChapter 6.3: Matrix Operations and Their Applications includes 87 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Since 87 problems in chapter 6.3: Matrix Operations and Their Applications have been answered, more than 37125 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9780321782281.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

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 ().

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.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.