Linear Algebra with Applications (2-Download)
5th Edition
ISBN: 9780321796974
Author: Otto Bretscher
Publisher: PEARSON
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Chapter 8, Problem 17E
To determine
To check: Whether the given statement is true or false.
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Chapter 8 Solutions
Linear Algebra with Applications (2-Download)
Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices in Exercises 1 through 6,...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - For each of the matrices A in Exercises 7 through...
Ch. 8.1 - For each of the matrices A in Exercises 7 through...Ch. 8.1 - Let L from R3 to R3 be the reflection about the...Ch. 8.1 - Consider a symmetric 33 matrix A with A2=I3 . Is...Ch. 8.1 - In Example 3 of this section, we diagonalized the...Ch. 8.1 - If A is invertible and orthogonally...Ch. 8.1 - Find the eigenvalues of the matrix...Ch. 8.1 - Use the approach of Exercise 16 to find the...Ch. 8.1 - Consider unit vector v1,...,vn in Rn such that the...Ch. 8.1 - Consider a linear transformation L from Rm to Rn ....Ch. 8.1 - Consider a linear transformation T from Rm to Rn ,...Ch. 8.1 - Consider a symmetric 33 matrix A with eigenvalues...Ch. 8.1 - Consider the matrix A=[0200k0200k0200k0] , where k...Ch. 8.1 - If an nn matrix A is both symmetric and...Ch. 8.1 - Consider the matrix A=[0001001001001000] . Find an...Ch. 8.1 - Consider the matrix [0000100010001000100010000] ....Ch. 8.1 - Let Jn be the nn matrix with all ones on the...Ch. 8.1 - Diagonalize the nn matrix (All ones along both...Ch. 8.1 - Diagonalize the 1313 matrix (All ones in the last...Ch. 8.1 - Consider a symmetric matrix A. If the vector v is...Ch. 8.1 - Consider an orthogonal matrix R whose first column...Ch. 8.1 - True or false? If A is a symmetric matrix, then...Ch. 8.1 - Consider the nn matrix with all ones on the main...Ch. 8.1 - For which angles(s) can you find three distinct...Ch. 8.1 - For which angles(s) can you find four distinct...Ch. 8.1 - Consider n+1 distinct unit vectors in Rn such that...Ch. 8.1 - Consider a symmetric nn matrix A with A2=A . Is...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 22 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - If A is any symmetric 33 matrix with eigenvalues...Ch. 8.1 - Show that for every symmetric nn matrix A, there...Ch. 8.1 - Find a symmetric 22 matrix B such that...Ch. 8.1 - For A=[ 2 11 11 11 2 11 11 11 2 ] find a nonzero...Ch. 8.1 - Consider an invertible symmetric nn matrix A. When...Ch. 8.1 - We say that an nnmatrix A is triangulizable if A...Ch. 8.1 - a. Consider a complex upper triangular nnmatrix U...Ch. 8.1 - Let us first introduce two notations. For a...Ch. 8.1 - Let U0 be a real upper triangular nn matrix with...Ch. 8.1 - Let R be a complex upper triangular nnmatrix with...Ch. 8.1 - Let A be a complex nnmatrix that ||1 for all...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - For each of the quadratic forms q listed in...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - Determine the definiteness of the quadratic forms...Ch. 8.2 - If A is a symmetric matrix, what can you say about...Ch. 8.2 - Recall that a real square matrix A is called skew...Ch. 8.2 - Consider a quadratic form q(x)=xAx on n and a...Ch. 8.2 - If A is an invertible symmetric matrix, what is...Ch. 8.2 - Show that a quadratic form q(x)=xAx of two...Ch. 8.2 - Show that the diagonal elements of a positive...Ch. 8.2 - Consider a 22 matrix A=[abbc] , where a and det A...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - Sketch the curves defined in Exercises 15 through...Ch. 8.2 - a. Sketch the following three surfaces:...Ch. 8.2 - On the surface x12+x22x32+10x1x3=1 find the two...Ch. 8.2 - Prob. 23ECh. 8.2 - Consider a quadratic form q(x)=xAx Where A is a...Ch. 8.2 - Prob. 25ECh. 8.2 - Prob. 26ECh. 8.2 - Consider a quadratic form q(x)=xAx , where A is a...Ch. 8.2 - Show that any positive definite nnmatrix A can be...Ch. 8.2 - For the matrix A=[8225] , write A=BBT as discussed...Ch. 8.2 - Show that any positive definite matrix A can be...Ch. 8.2 - Prob. 31ECh. 8.2 - Prob. 32ECh. 8.2 - Prob. 33ECh. 8.2 - Prob. 34ECh. 8.2 - Prob. 35ECh. 8.2 - Prob. 36ECh. 8.2 - Prob. 37ECh. 8.2 - Prob. 38ECh. 8.2 - Prob. 39ECh. 8.2 - Prob. 40ECh. 8.2 - Prob. 41ECh. 8.2 - Prob. 42ECh. 8.2 - Prob. 43ECh. 8.2 - Prob. 44ECh. 8.2 - Prob. 45ECh. 8.2 - Prob. 46ECh. 8.2 - Prob. 47ECh. 8.2 - Prob. 48ECh. 8.2 - Prob. 49ECh. 8.2 - Prob. 50ECh. 8.2 - What are the signs of the determinants of the...Ch. 8.2 - Consider a quadratic form q. If A is a symmetric...Ch. 8.2 - Consider a quadratic form q(x1,...,xn) with...Ch. 8.2 - If A is a positive semidefinite matrix with a11=0...Ch. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.2 - Prob. 57ECh. 8.2 - Prob. 58ECh. 8.2 - Prob. 59ECh. 8.2 - Prob. 60ECh. 8.2 - Prob. 61ECh. 8.2 - Prob. 62ECh. 8.2 - Prob. 63ECh. 8.2 - Prob. 64ECh. 8.2 - Prob. 65ECh. 8.2 - Prob. 66ECh. 8.2 - Prob. 67ECh. 8.2 - Prob. 68ECh. 8.2 - Prob. 69ECh. 8.2 - Prob. 70ECh. 8.2 - Prob. 71ECh. 8.3 - Find the singular values of A=[1002] .Ch. 8.3 - Let A be an orthogonal 22 matrix. Use the image of...Ch. 8.3 - Let A be an orthogonal nn matrix. Find the...Ch. 8.3 - Find the singular values of A=[1101] .Ch. 8.3 - Find the singular values of A=[pqqp] . Explain...Ch. 8.3 - Prob. 6ECh. 8.3 - Prob. 7ECh. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - Find singular value decompositions for the...Ch. 8.3 - If A is an invertible 22 matrix, what is the...Ch. 8.3 - If A is an invertible nn matrix, what is the...Ch. 8.3 - Consider an nm matrix A with rank(A)=m , and a...Ch. 8.3 - Prob. 18ECh. 8.3 - Prob. 19ECh. 8.3 - Prob. 20ECh. 8.3 - Prob. 21ECh. 8.3 - Consider the standard matrix A representing the...Ch. 8.3 - Consider an SVD A=UVT of an nm matrix A. Show that...Ch. 8.3 - If A is a symmetric nn matrix, what is the...Ch. 8.3 - Prob. 25ECh. 8.3 - Prob. 26ECh. 8.3 - Prob. 27ECh. 8.3 - Prob. 28ECh. 8.3 - Prob. 29ECh. 8.3 - Prob. 30ECh. 8.3 - Show that any matrix of rank r can be written as...Ch. 8.3 - Prob. 32ECh. 8.3 - Prob. 33ECh. 8.3 - For which square matrices A is there a singular...Ch. 8.3 - Prob. 35ECh. 8.3 - Prob. 36ECh. 8 - The singular values of any diagonal matrix D are...Ch. 8 - Prob. 2ECh. 8 - Prob. 3ECh. 8 - Prob. 4ECh. 8 - Prob. 5ECh. 8 - Prob. 6ECh. 8 - The function q(x1,x2)=3x12+4x1x2+5x2 is a...Ch. 8 - Prob. 8ECh. 8 - If matrix A is positive definite, then all the...Ch. 8 - Prob. 10ECh. 8 - Prob. 11ECh. 8 - Prob. 12ECh. 8 - Prob. 13ECh. 8 - Prob. 14ECh. 8 - Prob. 15ECh. 8 - Prob. 16ECh. 8 - Prob. 17ECh. 8 - Prob. 18ECh. 8 - Prob. 19ECh. 8 - Prob. 20ECh. 8 - Prob. 21ECh. 8 - Prob. 22ECh. 8 - If A and S are invertible nn matrices, then...Ch. 8 - Prob. 24ECh. 8 - Prob. 25ECh. 8 - Prob. 26ECh. 8 - Prob. 27ECh. 8 - Prob. 28ECh. 8 - Prob. 29ECh. 8 - Prob. 30ECh. 8 - Prob. 31ECh. 8 - Prob. 32ECh. 8 - Prob. 33ECh. 8 - Prob. 34ECh. 8 - Prob. 35ECh. 8 - Prob. 36ECh. 8 - Prob. 37ECh. 8 - Prob. 38ECh. 8 - Prob. 39ECh. 8 - Prob. 40ECh. 8 - Prob. 41ECh. 8 - Prob. 42ECh. 8 - Prob. 43ECh. 8 - Prob. 44ECh. 8 - Prob. 45ECh. 8 - Prob. 46ECh. 8 - Prob. 47ECh. 8 - Prob. 48ECh. 8 - Prob. 49ECh. 8 - Prob. 50ECh. 8 - Prob. 51ECh. 8 - Prob. 52ECh. 8 - Prob. 53ECh. 8 - Prob. 54E
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- Can a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 22 matrix.arrow_forwardCan a matrix with an entire column of zeros have an inverse? Explain why or why not.arrow_forwardDetermine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.arrow_forward
- Let A be an nn matrix in which the entries of each row sum to zero. Find |A|.arrow_forwardIn general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. In Exercises 38-41, show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that .arrow_forwardFind two nonzero matrices A and B such that AB=BA.arrow_forward
- Determine if the statement is true or false. If the statement is false, then correct it and make it true. Every matrix has a unique reduced row-echelon form.arrow_forwardExplain what it means in terms of an inverse for a matrix to have a 0 determinant.arrow_forwardShow that no 22 matrices A and B exist that satisfy the matrix equation. AB-BA=1001.arrow_forward
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