25. a. Show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros. b. Find a similar result involving a column of zeros. 26. In each part, find a 6 × 6 matrix [a] that satisfies the stated condition. Make your answers as general as possible by using letters rather than specific numbers for the nonzero entries. a. aij = 0 if if i #j ij if |i-j|>1
25. a. Show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros. b. Find a similar result involving a column of zeros. 26. In each part, find a 6 × 6 matrix [a] that satisfies the stated condition. Make your answers as general as possible by using letters rather than specific numbers for the nonzero entries. a. aij = 0 if if i #j ij if |i-j|>1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Matrix Problems and Solutions
#### 25. Matrix Operations with Rows and Columns of Zeros
**a. Show that if \(A\) has a row of zeros and \(B\) is any matrix for which \(AB\) is defined, then \(AB\) also has a row of zeros.**
To demonstrate this, consider:
- \(A\) as an \(m \times n\) matrix.
- \(B\) as an \(n \times p\) matrix.
Assume the \(i\)th row of \(A\) is a row of zeros, i.e., \(\mathbf{a}_i = [0, 0, \ldots, 0]\). Then the \(i\)th row of the product \(AB\) is given by \(\mathbf{a}_i B = [0, 0, \ldots, 0] B = [0, 0, \ldots, 0]\). Therefore, the \(i\)th row of \(AB\) is a row of zeros.
**b. Find a similar result involving a column of zeros.**
Consider:
- \(A\) as an \(m \times n\) matrix.
- \(B\) as an \(n \times p\) matrix.
Assume the \(j\)th column of \(B\) is a column of zeros, i.e., \(\mathbf{b}_j = [0, 0, \ldots, 0]^T\). Then the product of \(A\) with the \(j\)th column of \(B\) results in the \(j\)th column of \(AB\) being a column of zeros, because:
\[
AB = [A\mathbf{b}_1, A\mathbf{b}_2, \ldots, A\mathbf{b}_p]
\]
where each \(\mathbf{b}_k\) is a column vector. If \(\mathbf{b}_j = [0, 0, \ldots, 0]^T\), then \(A \mathbf{b}_j = [0, 0, \ldots, 0]^T\), contributing a zero column to \(AB\).
#### 26. Constructing a \(6 \times 6\) Matrix with Specific Conditions
**In each part, find](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2Ff629b490-c01f-48b8-b0a9-a8597f6b9d50%2Fd4pmpb_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Problems and Solutions
#### 25. Matrix Operations with Rows and Columns of Zeros
**a. Show that if \(A\) has a row of zeros and \(B\) is any matrix for which \(AB\) is defined, then \(AB\) also has a row of zeros.**
To demonstrate this, consider:
- \(A\) as an \(m \times n\) matrix.
- \(B\) as an \(n \times p\) matrix.
Assume the \(i\)th row of \(A\) is a row of zeros, i.e., \(\mathbf{a}_i = [0, 0, \ldots, 0]\). Then the \(i\)th row of the product \(AB\) is given by \(\mathbf{a}_i B = [0, 0, \ldots, 0] B = [0, 0, \ldots, 0]\). Therefore, the \(i\)th row of \(AB\) is a row of zeros.
**b. Find a similar result involving a column of zeros.**
Consider:
- \(A\) as an \(m \times n\) matrix.
- \(B\) as an \(n \times p\) matrix.
Assume the \(j\)th column of \(B\) is a column of zeros, i.e., \(\mathbf{b}_j = [0, 0, \ldots, 0]^T\). Then the product of \(A\) with the \(j\)th column of \(B\) results in the \(j\)th column of \(AB\) being a column of zeros, because:
\[
AB = [A\mathbf{b}_1, A\mathbf{b}_2, \ldots, A\mathbf{b}_p]
\]
where each \(\mathbf{b}_k\) is a column vector. If \(\mathbf{b}_j = [0, 0, \ldots, 0]^T\), then \(A \mathbf{b}_j = [0, 0, \ldots, 0]^T\), contributing a zero column to \(AB\).
#### 26. Constructing a \(6 \times 6\) Matrix with Specific Conditions
**In each part, find
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