 10.2.1: Two matrices are ________ if all of their corresponding entries are...
 10.2.2: When performing matrix operations, real numbers are often referred ...
 10.2.3: A matrix consisting entirely of zeros is called a ________ matrix a...
 10.2.4: The matrix consisting of 1s on its main diagonal and 0s elsewhere i...
 10.2.5: (a) (i) Distributive Property (b) (ii) Commutative Property of Matr...
 10.2.6: (a) (i) Distributive Property (b) (ii) Additive Identity of Matrix ...
 10.2.7:
 10.2.8: 5
 10.2.9: 16 3 0 4 13 2 5 15 4 4 6 0
 10.2.10: x 2 1 7 8 2y 2 3 2x y 2 2x 6 1 7 8 18 2 3 8 11 1
 10.2.11: A 1 2 1 1,
 10.2.12: B 3 4 2 2 A 1 2 2 1,
 10.2.13: A 8 2 4 1 3 5 , B
 10.2.14: A 1 0 1 6 3 9,
 10.2.15: A 4 1 5 2 1 2 3 1 4 0, B
 10.2.16: A 1 3 5 0 4 4 2 4 8 1 0 2 1 6 0 , B
 10.2.17: A 6 1 0 4 3 0,
 10.2.18: A B 462 3 2 1 ,
 10.2.19: 5 3 0 6 7 2 1 1 10 14 8 6 A B
 10.2.20: 6 1 8 0 0 3 5 1 11 2 7 1 5
 10.2.21: 4 4 0 0 2 1 3 2 3 1 6 2 0 6
 10.2.22:
 10.2.23:
 10.2.24:
 10.2.25: 3 7 2 1 5 4 6 3 2 0 2
 10.2.26: 55 14 22 11 19 22 13 20 6 3
 10.2.27: 3.211 1.004 0.055 6.829 4.914 3.889 1.630 5.256 9.768 3.090 8.335 4...
 10.2.28: 10 20 12 15 10 4 1 8 13 7 6 11 0 9 3 3 14 13 8 15 3.2
 10.2.29: X 3A 2B 2
 10.2.30: 2X 2A B
 10.2.31: 2X 3A B
 10.2.32: 2A 4B 2X
 10.2.33: A 2 3 1 1 4 6 , B 0 4 8 1 0 1 0 2 7 AB
 10.2.34: A 0 6 7 1 0 1 2 3 8 , B 2 4 1 1 5 6 A
 10.2.35: A 1 4 0 6 5 3 , B 2 0 3 9 A
 10.2.36: A 1 0 0 0 4 0 0 0 2 , B 3 0 0 0 1 0 0 0 5 A
 10.2.37: A 5 0 0 0 8 0 0 0 7 , B 1 5 0 0 0 1 8 0 0 0 1 2 A
 10.2.38: A 0 0 0 0 0 0 5 3 4 , B 6 8 0 11 16 0 4 4 0 A
 10.2.39: A B 6 216
 10.2.40: A 1 6 0 13 3 8 2 17,
 10.2.41: A 7 2 10 5 5 4 4 1 7 , B 2 8 4 2 1 2 3 4 8 AB, B
 10.2.42: A 11 14 6 12 10 2 4 12 9 , B 12 5 15 10 12 16 A
 10.2.43: A 3 12 5 8 15 1 6 9 1 8 6 5 , A
 10.2.44: A 2 21 13 4 5 2 8 6 6 , B 2 7 32 0.5 0 15 14 1.6 B
 10.2.45: A 9 100 10 50 38 250 18 75, A
 10.2.46: A 16 4 9 18 13 21 ,
 10.2.47: A 1 4 2 2,
 10.2.48: A 6 2 3 4, B 2 2 0 4 B
 10.2.49: A 3 1 1 3,
 10.2.50: A 1 1 1 1,
 10.2.51: A B 112 7 8 1 ,
 10.2.52: A 321 , B 2 3 0
 10.2.53:
 10.2.54:
 10.2.55: 0 4 2 1 2 2 4 0 1 0 1 2 2 3 0 3 5 3 3
 10.2.56: 3 1 5 7 5 6 7 1 8 9
 10.2.57: 10 x1 2x1 x2 x2 4 0 X
 10.2.58: 2x1 x1 3x2 4x2 5 10
 10.2.59: 2x1 6x1 3x2 x2 4 36
 10.2.60: 4x1 x1 9x2 3x2 13 12 2x
 10.2.61: x1 2x2 3x3 x1 3x2 x3 2x1 5x2 5x3 9 6 17 4
 10.2.62: x1 x1 x1 x2 2x2 x2 3x3 x3 1 1 2
 10.2.63: x1 3x1 5x2 x2 2x2 2x3 x3 5x3 20 8 16 x1
 10.2.64: x1 x1 x2 3x2 6x2 4x3 5x3 17 11 40
 10.2.65: MANUFACTURING A corporation has three factories, each of which manu...
 10.2.66: MANUFACTURING A corporation has four factories, each of which manuf...
 10.2.67: AGRICULTURE A fruit grower raises two crops, apples and peaches. Ea...
 10.2.68: REVENUE An electronics manufacturer produces three models of LCD te...
 10.2.69: INVENTORY A company sells five models of computers through three re...
 10.2.70: VOTING PREFERENCES The matrix From RD I is called a stochastic matr...
 10.2.71: VOTING PREFERENCES Use a graphing utility to find and for the matri...
 10.2.72: LABOR/WAGE REQUIREMENTS A company that manufactures boats has the f...
 10.2.73: PROFIT At a local dairy mart, the numbers of gallons of skim milk, ...
 10.2.74: PROFIT At a convenience store, the numbers of gallons of 87octane,...
 10.2.75: EXERCISE The numbers of calories burned by individuals of different...
 10.2.76: HEALTH CARE The health care plans offered this year by a local manu...
 10.2.77: Two matrices can be added only if they have the same order.
 10.2.78: Matrix multiplication is commutative.
 10.2.79: A 2C
 10.2.80: B 3C
 10.2.81: AB
 10.2.82: BC
 10.2.83: BC D C
 10.2.84: CB D
 10.2.85: D A 3B B
 10.2.86: BC DA
 10.2.87: Consider matrices and below. Perform the indicated operations and c...
 10.2.88: Use the following matrices to find and What do your results tell yo...
 10.2.89: THINK ABOUT IT If and are real numbers such that and then However, ...
 10.2.90: THINK ABOUT IT If and are real numbers such that then or However, i...
 10.2.91: Let and be unequal diagonal matrices of the same order. (A diagonal...
 10.2.92: Let and let and (a) Find and Identify any similarities with and (b)...
 10.2.93: Find two matrices and such that BA AB BA.
 10.2.94: CAPSTONE Let matrices and be of orders and respectively. Answer the...
Solutions for Chapter 10.2: Operations with Matrices
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 10.2: Operations with Matrices
Get Full SolutionsSince 94 problems in chapter 10.2: Operations with Matrices have been answered, more than 49319 students have viewed full stepbystep solutions from this chapter. Chapter 10.2: Operations with Matrices includes 94 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474.

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

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).