Compute ATBT and BT AT by hand if possible. If not possible, describe why. A = 2 3 −1 0 B = 10 -1 0 9 -1 0
Compute ATBT and BT AT by hand if possible. If not possible, describe why. A = 2 3 −1 0 B = 10 -1 0 9 -1 0
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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![To compute \( A^T B^T \) and \( B^T A^T \) by hand, we are given the matrices:
\[ A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} \]
\[ B = \begin{bmatrix} 10 & 0 & 9 \\ -1 & -1 & 0 \end{bmatrix} \]
### Transpose of Matrices
The transpose of a matrix is obtained by swapping its rows and columns.
- **Transpose of A (\( A^T \)):**
\[ A^T = \begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix} \]
- **Transpose of B (\( B^T \)):**
\[ B^T = \begin{bmatrix} 10 & -1 \\ 0 & -1 \\ 9 & 0 \end{bmatrix} \]
### Matrix Multiplication
#### Calculating \( A^T B^T \)
- **Dimensions of \( A^T \):** \( 2 \times 2 \)
- **Dimensions of \( B^T \):** \( 3 \times 2 \)
The matrix multiplication \( A^T B^T \) is not possible because the number of columns in \( A^T \) does not match the number of rows in \( B^T \).
#### Calculating \( B^T A^T \)
- **Dimensions of \( B^T \):** \( 3 \times 2 \)
- **Dimensions of \( A^T \):** \( 2 \times 2 \)
The resulting matrix from \( B^T A^T \) will be of dimension \( 3 \times 2 \), and can be computed as follows:
1. **First row of \( B^T \) and columns of \( A^T \):**
\[ (10 \times 2) + (-1 \times -1) = 20 + 1 = 21 \]
\[ (10 \times 3) + (-1 \times 0) = 30 + 0 = 30 \]
2. **Second row of \( B^T \) and columns of \( A^T \):**
\[ (0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F182bf40e-0e0c-42ab-83b6-0330feba0dfc%2Fdx8kmcnx_processed.png&w=3840&q=75)
Transcribed Image Text:To compute \( A^T B^T \) and \( B^T A^T \) by hand, we are given the matrices:
\[ A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} \]
\[ B = \begin{bmatrix} 10 & 0 & 9 \\ -1 & -1 & 0 \end{bmatrix} \]
### Transpose of Matrices
The transpose of a matrix is obtained by swapping its rows and columns.
- **Transpose of A (\( A^T \)):**
\[ A^T = \begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix} \]
- **Transpose of B (\( B^T \)):**
\[ B^T = \begin{bmatrix} 10 & -1 \\ 0 & -1 \\ 9 & 0 \end{bmatrix} \]
### Matrix Multiplication
#### Calculating \( A^T B^T \)
- **Dimensions of \( A^T \):** \( 2 \times 2 \)
- **Dimensions of \( B^T \):** \( 3 \times 2 \)
The matrix multiplication \( A^T B^T \) is not possible because the number of columns in \( A^T \) does not match the number of rows in \( B^T \).
#### Calculating \( B^T A^T \)
- **Dimensions of \( B^T \):** \( 3 \times 2 \)
- **Dimensions of \( A^T \):** \( 2 \times 2 \)
The resulting matrix from \( B^T A^T \) will be of dimension \( 3 \times 2 \), and can be computed as follows:
1. **First row of \( B^T \) and columns of \( A^T \):**
\[ (10 \times 2) + (-1 \times -1) = 20 + 1 = 21 \]
\[ (10 \times 3) + (-1 \times 0) = 30 + 0 = 30 \]
2. **Second row of \( B^T \) and columns of \( A^T \):**
\[ (0
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