Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
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Chapter 1.4, Problem 20E

Let A be an n × n matrix. Show that if A k + 1 = O , then I − A is nonsingular and

( I − A ) − 1 = I + A + A 2 + ... A k

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Chapter 1 Solutions

Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))

Ch. 1.1 - Use back substitution to solve each of the...Ch. 1.1 - Write out the coefficient matrix for each of the...Ch. 1.1 - In each of the following systems, interpret each...Ch. 1.1 - Write an augmented matrix for each of the systems...Ch. 1.1 - Write out the system of equations that corresponds...Ch. 1.1 - Solve each of the following systems. (a)...Ch. 1.1 - The two systems 2x1+x2=34x1+3x2=5 and...Ch. 1.1 - Solve the two systems...Ch. 1.1 - Given a System of the form m1x1+x2=b1m2x1+x2=b2...Ch. 1.1 - Consider a system of the form...
Ch. 1.1 - Give a geometrical interpretation of a linear...Ch. 1.2 - Which of the matrices that follow are in row...Ch. 1.2 - The augmented matrices that follow are in row...Ch. 1.2 - The augmented matrices that follow are in reduced...Ch. 1.2 - For each of the systems in Exercise 3, make a list...Ch. 1.2 - For each of the systems of equations that follow,...Ch. 1.2 - Use GaussJordan reduction to solve each of the...Ch. 1.2 - Give a geometric explanation of why a homogeneous...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Consider a linear system whose augmented matrix is...Ch. 1.2 - Given the linear systems...Ch. 1.2 - Given the linear systems (i)...Ch. 1.2 - Given a homogeneous system of linear equations, if...Ch. 1.2 - Given a nonhomogeneous system of linear equations,...Ch. 1.2 - Determine the values ofx1,x2,x3,x4for the...Ch. 1.2 - Prob. 16ECh. 1.2 - Prob. 17ECh. 1.2 - In Application 3 the solution (6, 6, 6, 1) was...Ch. 1.2 - Prob. 19ECh. 1.2 - Nitric acid is prepared commercially by a series...Ch. 1.2 - Prob. 21ECh. 1.2 - Prob. 22ECh. 1.3 - If A=[314201122]andB=[102311241] compute (a) 2A...Ch. 1.3 - For each of the pairs of matrices that follow,...Ch. 1.3 - For which of the pairs in Exercise 2 is it...Ch. 1.3 - Write each of the following systems of equations...Ch. 1.3 - If A=[341127] verify that (a) 5A=3A+2A (b)...Ch. 1.3 - If A=[ 4 2 1 3 6 5 ]andB=[ 1 2 3 2 2 4 ] verify...Ch. 1.3 - If A=[216324]andB=[2416] verify that (a)...Ch. 1.3 - If A=[ 2 1 4 3 ],B=[ 2 0 1 4 ],C=[ 3 2 1 1 ]...Ch. 1.3 - Let A=[ 1 1 2 2 ],b=[40],c=[32] (a) Write b as a...Ch. 1.3 - For each of the choices of A and b that follow,...Ch. 1.3 - Let Abe a 53 matrix. If b=a1+a2=a2+a3 then what...Ch. 1.3 - Let Abe a 34 matrix. If b=a1+a2+a3+a4 then what...Ch. 1.3 - Let Ax=b be a linear system whose augmented matrix...Ch. 1.3 - Prob. 14ECh. 1.3 - Let A be an mn matrix. Explain why the matrix...Ch. 1.3 - A matrix A is said to be skew symmetric if AT=A ....Ch. 1.3 - In Application 3, suppose that we are searching...Ch. 1.3 - Let A be a 22 matrix with a110 and let =a21/a11 ....Ch. 1.4 - Explain why each of the following algebraic rules...Ch. 1.4 - Will the rules in Exercise 1 work if a is replaced...Ch. 1.4 - Find nonzero 22 matrices A and B such that AB=0 .Ch. 1.4 - Find nonzero matrices A, B, and C such that...Ch. 1.4 - The matrix A=[1111] has the property that A2=O ....Ch. 1.4 - Prove the associative law of multiplication for 22...Ch. 1.4 - Let A=[ 1 2 1 2 1 2 1 2] Compute A2 and A3 . What...Ch. 1.4 - Let A=[ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1...Ch. 1.4 - Let A=[0100001000010000] Show that An=O for n4 .Ch. 1.4 - Let A and B be symmetric nn matrices. For each of...Ch. 1.4 - Let C be nonsymmetric nn matrix. For each of the...Ch. 1.4 - Let A=[ a 11 a 12 a 21 a 22] Show that if...Ch. 1.4 - Use the result from Exercise 12 to find the...Ch. 1.4 - Let A and B are nn matrices. Show that if...Ch. 1.4 - Let A be a nonsingular matrix. Show that A1 is...Ch. 1.4 - Prove that if A is nonsingular then AT is...Ch. 1.4 - Let A be an nn matrix and let x and y be vectors...Ch. 1.4 - Let A be a nonsingular nn matrix. Use mathematical...Ch. 1.4 - Let A be an nn matrix. Show that if A2=O , then IA...Ch. 1.4 - Let A be an nn matrix. Show that if Ak+1=O , then...Ch. 1.4 - Given R=[cossinsincos] Show that R is nonsingular...Ch. 1.4 - An nn matrix A is said to be an involutionifA2=I ....Ch. 1.4 - Let u be a unity vector in n (i.e. uTu=1 ) and let...Ch. 1.4 - A matrix A is said to be an idempotentif A2=A ....Ch. 1.4 - Prob. 25ECh. 1.4 - Let D be an nn diagonal matrix whose diagonal...Ch. 1.4 - Let Abe an involution matrix and let...Ch. 1.4 - Let A be an mn matrix. Show that ATA and AAT are...Ch. 1.4 - Let A and B be symmetric nn matrices. Prove that...Ch. 1.4 - Let Abe an nn matrix and let B=A+ATandC=AAT (a)...Ch. 1.4 - In Application 1, how many married women and how...Ch. 1.4 - Consider the matrix A=[ 0 1 0 1 1 1 0 1 1 0 0 1 0...Ch. 1.4 - Consider the graph (a) Determine the adjacency...Ch. 1.4 - If Ax=Bx for some nonzero vector x, then the...Ch. 1.4 - If A and B are singular nn matrices, then A+B is...Ch. 1.4 - If A and B are nonsingular matrices, then (AB)T is...Ch. 1.5 - Which of the matrices that follow are elementary...Ch. 1.5 - Find the inverse of each matrix in Exercise 1. For...Ch. 1.5 - Prob. 3ECh. 1.5 - Prob. 4ECh. 1.5 - Let A=[ 1 2 1 2 1 0 4 3 2 ],B=[ 1 2 2 2 1 2 4 3 6...Ch. 1.5 - Prob. 6ECh. 1.5 - Let A=[2164] (a) Express A1 as a product of...Ch. 1.5 - Compute the LU factorization of each of the...Ch. 1.5 - Let A=[ 1 3 2 0 3 2 1 4 3 ] (a) Verify that A1=[ 1...Ch. 1.5 - Find the inverse of each of the following...Ch. 1.5 - Prob. 11ECh. 1.5 - Let A=[ 5 3 3 2 ],B=[ 6 2 2 4 ],C=[ 4 6 2 3 ]...Ch. 1.5 - Is the transpose of an elementary matrix an...Ch. 1.5 - Let U and R bennupper triangular matrices and...Ch. 1.5 - Let A be a 33 matrix and suppose that 2a1+a24a3=0...Ch. 1.5 - Prob. 16ECh. 1.5 - Let A and B be nn matrices and let C=AB . Show...Ch. 1.5 - Prob. 18ECh. 1.5 - Let U be an nn upper triangular matrix with...Ch. 1.5 - Prob. 20ECh. 1.5 - Prob. 21ECh. 1.5 - Show that if A is a symmetric nonsingular matrix...Ch. 1.5 - Prove that if A is a row equivalent to B then B is...Ch. 1.5 - Prob. 24ECh. 1.5 - Prob. 25ECh. 1.5 - Prove that B is row equivalent to A if and only if...Ch. 1.5 - Is it possible for a singular matrix B to be row...Ch. 1.5 - Prob. 28ECh. 1.5 - Prob. 29ECh. 1.5 - Prob. 30ECh. 1.5 - Prob. 31ECh. 1.5 - Prob. 32ECh. 1.6 - Let A be a nonsingular nn matrix. Perform the...Ch. 1.6 - Prob. 2ECh. 1.6 - Let A=[ 1 2 1 1 ]andB=[ 2 1 1 3 ] (a) Calculate...Ch. 1.6 - Let I=[ 1 0 0 1 ],E=[ 0 1 1 0 ],O=[ 0 0 0 0 ] C=[...Ch. 1.6 - Perform each of the following block...Ch. 1.6 - Given X=[ 2 4 1 2 5 3 ],Y=[ 1 2 2 3 4 1 ] (a)...Ch. 1.6 - Let A=[ A 11 A 21 A 12 A 22 ]andAT=[ A 11 T A 12 T...Ch. 1.6 - Let Abe an mn matrix, X and nr matrix, and B an mn...Ch. 1.6 - Prob. 9ECh. 1.6 - Prob. 10ECh. 1.6 - Prob. 11ECh. 1.6 - Let A and B be nn matrices and let M be a block...Ch. 1.6 - Prob. 13ECh. 1.6 - Prob. 14ECh. 1.6 - Prob. 15ECh. 1.6 - Prob. 16ECh. 1.6 - Prob. 17ECh. 1.6 - Let A, B, L, M, S, and T be nn matrices with A, B,...Ch. 1.6 - Let Abe an nn matrix and xn . (a) A scalar c can...Ch. 1.6 - If A is an nn matrix with the property that Ax=0...Ch. 1.6 - Prob. 21ECh. 1.6 - Prob. 22ECh. 1 - Use MATLAB to generate random 44 matrices A and B....Ch. 1 - Set n=200 and generate an nn matrix and two...Ch. 1 - Set A=floor(10*rand(6)) . By construction, the...Ch. 1 - Construct a mainx as follows: Set...Ch. 1 - Generate a matrix A by setting A = floor(10 *...Ch. 1 - Consider the graph (a) Determine the adjacency...Ch. 1 - In Application 1 of Section 1.4, the numbers of...Ch. 1 - The following table describes a seven-stage model...Ch. 1 - Set A = magic(8) and then compute its reduced row...Ch. 1 - Set B=[1,1;1,1] and A=[zeros(2),eye(2);eye(2),B]...Ch. 1 - The MATLAB commands A = floor(10 * rand((6)), B=AA...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - This chapter test consists of trueorfalse...Ch. 1 - Find all solutions of the linear system...Ch. 1 - (a) A linear equation in two unknowns corresponds...Ch. 1 - LetAx=bbe a system of n linear equations in n...Ch. 1 - LetAbeamatrix of the form A=[22] where and are...Ch. 1 - Let A=[213427135],B=[213135427],C=[013027535] Find...Ch. 1 - Let A be a 33 matrix and let b=3a1+a2+4a3 Will the...Ch. 1 - Let A be a 33 matrix and suppose that a13a2+2a3=0...Ch. 1 - Given the vector x0=[11] is it possible to find 22...Ch. 1 - Let A and B be symmetric nn matrices and let C=AB...Ch. 1 - Let E and F be nn elementary matrices and let C=EF...Ch. 1 - Given A=[IOOOIOOBI] where all the submatrices are...Ch. 1 - LetA and B be 1010 matrices that are...

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