 2.1.2.1.1: Equality of Matrices In Exercises 14, find x and y.[x72y] = [47222]
 2.1.1: Equality of Matrices In Exercises 14, find x and y.[x72y] = [47222]
 2.1.2: Equality of Matrices In Exercises 14, find x and y.[5yx8] = [512138]
 2.1.2.1.2: Equality of Matrices In Exercises 14, find x and y.[5yx8] = [512138]
 2.1.3: Equality of Matrices In Exercises 14, find x and y.[163041325154460...
 2.1.2.1.3: Equality of Matrices In Exercises 14, find x and y.[163041325154460...
 2.1.4: Equality of Matrices In Exercises 14, find x and y.[x + 21782y232xy...
 2.1.2.1.4: Equality of Matrices In Exercises 14, find x and y.[x + 21782y232xy...
 2.1.5: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.5: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.6: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.6: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.7: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.7: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.8: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.8: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.9: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.9: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.10: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.2.1.10: Operations with Matrices In Exercises 510, find, if possible, (a) A...
 2.1.11: Find (a) c21 and (b) c13, where C = 2A 3B,A = [ 534142], and B = [1...
 2.1.2.1.11: Find (a) c21 and (b) c13, where C = 2A 3B,A = [ 534142], and B = [1...
 2.1.12: Find (a) c23 and (b) c32, where C = 5A + 2B,A = [ 4031131921], and ...
 2.1.2.1.12: Find (a) c23 and (b) c32, where C = 5A + 2B,A = [ 4031131921], and ...
 2.1.13: Solve for x, y, and z in the matrix equation4[xzy1] = 2[ yxz1] + 2[...
 2.1.2.1.13: Solve for x, y, and z in the matrix equation4[xzy1] = 2[ yxz1] + 2[...
 2.1.14: Solve for x, y, z, and w in the matrix equation[wyxx] = [4231] + 2[...
 2.1.2.1.14: Solve for x, y, z, and w in the matrix equation[wyxx] = [4231] + 2[...
 2.1.15: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.15: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.16: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.16: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.17: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.17: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.18: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.18: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.19: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.19: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.20: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.20: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.21: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.21: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.22: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.22: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.23: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.23: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.24: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.24: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.25: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.25: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.26: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.26: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.27: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.27: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.28: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.2.1.28: Finding Products of Two Matrices In Exercises 1528, find, if possib...
 2.1.29: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.29: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.30: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.30: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.31: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.31: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.32: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.32: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.33: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.33: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.34: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.34: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.35: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.35: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.36: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.2.1.36: Matrix Size In Exercises 2936, let A, B, C, D, and Ebe matrices wit...
 2.1.37: Solving a Matrix Equation In Exercises 37 and 38, solve the matrix ...
 2.1.2.1.37: Solving a Matrix Equation In Exercises 37 and 38, solve the matrix ...
 2.1.38: Solving a Matrix Equation In Exercises 37 and 38, solve the matrix ...
 2.1.2.1.38: Solving a Matrix Equation In Exercises 37 and 38, solve the matrix ...
 2.1.39: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.39: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.40: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.40: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.41: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.41: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.42: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.42: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.43: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.43: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.44: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.44: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.45: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.45: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.46: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.46: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.47: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.47: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.48: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.2.1.48: Solving a System of Linear Equations In Exercises 3948, write the s...
 2.1.49: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.2.1.49: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.50: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.2.1.50: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.51: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.2.1.51: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.52: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.2.1.52: Writing a Linear Combination In Exercises 4952, write the column ma...
 2.1.53: Solving a Matrix Equation In Exercises 53 and 54, solve for A.[1325...
 2.1.2.1.53: Solving a Matrix Equation In Exercises 53 and 54, solve for A.[1325...
 2.1.54: Solving a Matrix Equation In Exercises 53 and 54, solve for A.[2312...
 2.1.2.1.54: Solving a Matrix Equation In Exercises 53 and 54, solve for A.[2312...
 2.1.55: Solving a Matrix Equation In Exercises 55 and 56, solve the matrix ...
 2.1.2.1.55: Solving a Matrix Equation In Exercises 55 and 56, solve the matrix ...
 2.1.56: Solving a Matrix Equation In Exercises 55 and 56, solve the matrix ...
 2.1.2.1.56: Solving a Matrix Equation In Exercises 55 and 56, solve the matrix ...
 2.1.57: Diagonal Matrix In Exercises 57 and 58, find the product AA for the...
 2.1.2.1.57: Diagonal Matrix In Exercises 57 and 58, find the product AA for the...
 2.1.58: Diagonal Matrix In Exercises 57 and 58, find the product AA for the...
 2.1.2.1.58: Diagonal Matrix In Exercises 57 and 58, find the product AA for the...
 2.1.59: Finding Products of Diagonal Matrices In Exercises 59 and 60, find ...
 2.1.2.1.59: Finding Products of Diagonal Matrices In Exercises 59 and 60, find ...
 2.1.60: Finding Products of Diagonal Matrices In Exercises 59 and 60, find ...
 2.1.2.1.60: Finding Products of Diagonal Matrices In Exercises 59 and 60, find ...
 2.1.61: Guided Proof Prove that if A and B are diagonalmatrices (of the sam...
 2.1.2.1.61: Guided Proof Prove that if A and B are diagonalmatrices (of the sam...
 2.1.62: Writing Let A and B be 3 3 matrices, where A isdiagonal.(a) Describ...
 2.1.2.1.62: Writing Let A and B be 3 3 matrices, where A isdiagonal.(a) Describ...
 2.1.63: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.2.1.63: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.64: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.2.1.64: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.65: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.2.1.65: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.66: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.2.1.66: Trace of a Matrix In Exercises 6366, find the trace of the matrix. ...
 2.1.67: Proof Prove that each statement is true when A and Bare square matr...
 2.1.2.1.67: Proof Prove that each statement is true when A and Bare square matr...
 2.1.68: Proof Prove that if A and B are square matrices oforder n, then Tr(...
 2.1.2.1.68: Proof Prove that if A and B are square matrices oforder n, then Tr(...
 2.1.69: Find conditions on w, x, y, and z such that AB = BA forthe matrices...
 2.1.2.1.69: Find conditions on w, x, y, and z such that AB = BA forthe matrices...
 2.1.70: Verify AB = BA for the matrices below.A = [cos sin sin cos ] and B ...
 2.1.2.1.70: Verify AB = BA for the matrices below.A = [cos sin sin cos ] and B ...
 2.1.71: Show that the matrix equation has no solution.[1111] A = [1001]
 2.1.2.1.71: Show that the matrix equation has no solution.[1111] A = [1001]
 2.1.72: Show that no 2 2 matrices A and B exist that satisfythe matrix equa...
 2.1.2.1.72: Show that no 2 2 matrices A and B exist that satisfythe matrix equa...
 2.1.73: Exploration Let i = 1 and letA = [i00i] and B = [0ii0].(a) Find A2,...
 2.1.2.1.73: Exploration Let i = 1 and letA = [i00i] and B = [0ii0].(a) Find A2,...
 2.1.74: Guided Proof Prove that if the product AB is a squarematrix, then t...
 2.1.2.1.74: Guided Proof Prove that if the product AB is a squarematrix, then t...
 2.1.75: Proof Prove that if both products AB and BA are defined, then AB an...
 2.1.2.1.75: Proof Prove that if both products AB and BA are defined, then AB an...
 2.1.76: Let A and B be matrices such that the product AB is defined. Show t...
 2.1.2.1.76: Let A and B be matrices such that the product AB is defined. Show t...
 2.1.77: Let A and B be n n matrices. Show that if the ith row of A has all ...
 2.1.2.1.77: Let A and B be n n matrices. Show that if the ith row of A has all ...
 2.1.78: CAPSTONE Let matrices A and B be ofsizes 3 2 and 2 2, respectively....
 2.1.2.1.78: CAPSTONE Let matrices A and B be ofsizes 3 2 and 2 2, respectively....
 2.1.79: Agriculture A fruit grower raises two crops, applesand peaches. The...
 2.1.2.1.79: Agriculture A fruit grower raises two crops, applesand peaches. The...
 2.1.80: Manufacturing A corporation has three factories,each of which manuf...
 2.1.2.1.80: Manufacturing A corporation has three factories,each of which manuf...
 2.1.81: Politics In the matrixeach entry pij (i j) represents the proportio...
 2.1.2.1.81: Politics In the matrixeach entry pij (i j) represents the proportio...
 2.1.82: Population The matrices show the numbers ofpeople (in thousands) wh...
 2.1.2.1.82: Population The matrices show the numbers ofpeople (in thousands) wh...
 2.1.83: Block Multiplication In Exercises 83 and 84, perform the block mult...
 2.1.2.1.83: Block Multiplication In Exercises 83 and 84, perform the block mult...
 2.1.84: Block Multiplication In Exercises 83 and 84, perform the block mult...
 2.1.2.1.84: Block Multiplication In Exercises 83 and 84, perform the block mult...
 2.1.85: True or False? In Exercises 85 and 86, determine whether each state...
 2.1.2.1.85: True or False? In Exercises 85 and 86, determine whether each state...
 2.1.86: True or False? In Exercises 85 and 86, determine whether each state...
 2.1.2.1.86: True or False? In Exercises 85 and 86, determine whether each state...
 2.1.87: Evaluating an Expression In Exercises 16, evaluate the expression
 2.1.2.1.87: Evaluating an Expression In Exercises 16, evaluate the expression
Solutions for Chapter 2.1: Operations with Matrices
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 2.1: Operations with Matrices
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.1: Operations with Matrices includes 174 full stepbystep solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Since 174 problems in chapter 2.1: Operations with Matrices have been answered, more than 47264 students have viewed full stepbystep solutions from this chapter.

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.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

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

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

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 •

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·