 8.4.1: A matrix that has the same number of rows as columns is called a(n)...
 8.4.2: True or False Matrix addition is commutative.
 8.4.3: To find the product AB of two matrices A and B, the number of in ma...
 8.4.4: True or False Matrix multiplication is commutative.
 8.4.5: Suppose that A is a square n by n matrix that is nonsingular. The m...
 8.4.6: If a matrix A has no inverse, it is called .
 8.4.7: True or False The identity matrix has properties similar to those o...
 8.4.8: If represents a matrix equation where A is a nonsingular matrix, th...
 8.4.9: A + B
 8.4.10: A  B
 8.4.11: 4A
 8.4.12: 3B
 8.4.13: 3A  2B
 8.4.14: 2A + 4B
 8.4.15: AC
 8.4.16: BC
 8.4.17: CA
 8.4.18: CB
 8.4.19: C1A + B2
 8.4.20: 1A + B2C
 8.4.21: AC  3I2
 8.4.22: CA + 5I3
 8.4.23: CA  CB
 8.4.24: AC + BC
 8.4.25: B 2 2 1 0R B2 146 3 132R
 8.4.26: B 4 1 2 1R B 661 0 254 1 R
 8.4.27: B 1 23 0 1 4R C 1 2 1 0 2 4 S
 8.4.28: C 1 1 3 2 0 5 S B 2 8 1 36 0R
 8.4.29: C 101 241 361 S C 1 3 6 2 8 1 S
 8.4.30: C 4 2 3 0 12 1 01 S C 2 6 1 1 0 2 S
 8.4.31: B R 2 1 1 1R
 8.4.32: B R 3 1 2 1 B R
 8.4.33: B R 6 5 2 2 B R
 8.4.34: B R a Z 0 4 1 6 2 B R
 8.4.35: B 2 1 a a B R a Z 0
 8.4.36: B S b 3 b 2 R b Z 0
 8.4.37: C S 1 1 1 0 2 1 2 3 0 B S
 8.4.38: C S 1 02 1 23 1 1 0 C S
 8.4.39: C S 11 1 3 2 1 31 2 C S
 8.4.40: C 3 31 1 21 2 1 1 C S
 8.4.41: b 2x + y = 8 x + y = 5
 8.4.42: b 3x  y = 8 2x + y = 4
 8.4.43: b 2x + y = 0 x + y = 5
 8.4.44: b 3x  y = 4 2x + y = 5
 8.4.45: b 6x + 5y = 7 2x + 2y = 2
 8.4.46: b 4x + y = 0 6x  2y = 14
 8.4.47: b 6x + 5y = 13 2x + 2y = 5
 8.4.48: b 4x + y = 5 6x  2y = 9
 8.4.49: b 2x + y = 3 ax + ay = a a Z 0
 8.4.50: b bx + 3y = 2b + 3 bx + 2y = 2b + 2 b Z 0
 8.4.51: c 2x + y = 7 a ax + ay = 5 a Z 0
 8.4.52: b bx + 3y = 14 bx + 2y = 10 b Z 0
 8.4.53: c x  y + z = 0 2y + z = 1 2x  3y = 5
 8.4.54: c x + 2z = 6 x + 2y + 3z = 5 x  y = 6
 8.4.55: d x  y + z = 2 2y + z = 2 2x  3y = 1 2
 8.4.56: d x + 2z = 2 x + 2y + 3z =  3 2 x  y = 2
 8.4.57: c x + y + z = 9 3x + 2y  z = 8 3x + y + 2z = 1
 8.4.58: c 3x + 3y + z = 8 x + 2y + z = 5 2x  y + z = 4
 8.4.59: e x + y + z = 2 3x + 2y  z = 7 3 3x + y + 2z = 10 3
 8.4.60: c 3x + 3y + z = 1 x + 2y + z = 0 2x  y + z = 4
 8.4.61: B 4 2 2 1R
 8.4.62: C 3 1 2 6 1 S
 8.4.63: B 15 3 10 2R
 8.4.64: B 3 0 4 0R
 8.4.65: C 3 1 1 1 4 7 125 S
 8.4.66: C 1 1 3 2 4 1 571 S
 8.4.67: C 25 61 12 18 2 4 8 35 21 S
 8.4.68: C 18 3 4 6 20 14 10 25 15 S
 8.4.69: D 44 21 18 6 2 10 15 5 21 12 12 4 8 16 4 9T
 8.4.70: D 16 22 3 5 21 17 4 8 2 8 27 20 5 15 3 10T
 8.4.71: c 25x + 61y  12z = 10 18x  12y + 7y = 9 3x + 4y  z = 12
 8.4.72: c 25x + 61y  12z = 15 18x  12y + 7z = 3 3x + 4y  z = 12
 8.4.73: c 25x + 61y  12z = 21 18x  12y + 7z = 7 3x + 4y  z = 2
 8.4.74: c 25x + 61y  12z = 25 18x  12y + 7z = 10 3x + 4y  z = 4
 8.4.75: b 2x + 3y = 11 5x + 7y = 24
 8.4.76: b 2x + 8y = 8 x + 7y = 13
 8.4.77: c x  2y + 4z = 2 3x + 5y  2z = 17 4x  3y = 22
 8.4.78: c 2x + 3y  z = 2 4x + 3z = 6 6y  2z = 2
 8.4.79: c 5x  y + 4z = 2 x + 5y  4z = 3 7x + 13y  4z = 17
 8.4.80: c 3x + 2y  z = 2 2x + y + 6z = 7 2x + 2y  14z = 17
 8.4.81: c 2x  3y + z = 4 3x + 2y  z = 3 5y + z = 6
 8.4.82: c 4x + 3y + 2z = 6 3x + y  z = 2 x + 9y + z = 6
 8.4.83: College Tuition Nikki and Joe take classes at a community college, ...
 8.4.84: School Loan Interest Jamal and Stephanie each have school loans iss...
 8.4.85: Computing the Cost of Production The Acme Steel Company is a produc...
 8.4.86: Computing Profit Rizza Ford has two locations, one in the city and ...
 8.4.87: Cryptography One method of encryption is to use a matrix to encrypt...
 8.4.88: Economic Mobility The relative income of a child (low, medium, or h...
 8.4.89: Consider the 2 by 2 square matrix If show that A is nonsingular and...
Solutions for Chapter 8.4: Matrix Algebra
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 8.4: Matrix Algebra
Get Full SolutionsChapter 8.4: Matrix Algebra includes 89 full stepbystep solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 89 problems in chapter 8.4: Matrix Algebra have been answered, more than 29636 students have viewed full stepbystep solutions from this chapter.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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!).

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

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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.