Consider the matrices 1 23 -2 4 3 7 9 8 A = : B= 0 1 4 %3D 560 Compute AB' and determinant of A
Consider the matrices 1 23 -2 4 3 7 9 8 A = : B= 0 1 4 %3D 560 Compute AB' and determinant of A
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 Operations: Multiplication and Determinant
#### Given Matrices:
Consider the matrices \( A \) and \( B \) defined as follows:
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\]
\[
B = \begin{bmatrix}
-2 & 4 & 3 \\
7 & 9 & 8
\end{bmatrix}
\]
#### Task:
1. Compute the product \( AB^T \):
- \( B^T \) denotes the transpose of matrix \( B \).
- Perform the matrix multiplication of \( A \) with \( B^T \).
2. Determine the determinant of matrix \( A \):
- Use standard methods for finding the determinant of a 3x3 matrix.
### Solution:
1. **Transpose of Matrix \( B \):**
\[
B^T = \begin{bmatrix}
-2 & 7 \\
4 & 9 \\
3 & 8
\end{bmatrix}
\]
2. **Matrix Multiplication \( AB^T \):**
\[
AB^T = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\begin{bmatrix}
-2 & 7 \\
4 & 9 \\
3 & 8
\end{bmatrix}
\]
3. **Determinant of Matrix \( A \):**
\[
\text{det}(A) = \begin{vmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{vmatrix}
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Transcribed Image Text:### Matrix Operations: Multiplication and Determinant
#### Given Matrices:
Consider the matrices \( A \) and \( B \) defined as follows:
\[
A = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\]
\[
B = \begin{bmatrix}
-2 & 4 & 3 \\
7 & 9 & 8
\end{bmatrix}
\]
#### Task:
1. Compute the product \( AB^T \):
- \( B^T \) denotes the transpose of matrix \( B \).
- Perform the matrix multiplication of \( A \) with \( B^T \).
2. Determine the determinant of matrix \( A \):
- Use standard methods for finding the determinant of a 3x3 matrix.
### Solution:
1. **Transpose of Matrix \( B \):**
\[
B^T = \begin{bmatrix}
-2 & 7 \\
4 & 9 \\
3 & 8
\end{bmatrix}
\]
2. **Matrix Multiplication \( AB^T \):**
\[
AB^T = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{bmatrix}
\begin{bmatrix}
-2 & 7 \\
4 & 9 \\
3 & 8
\end{bmatrix}
\]
3. **Determinant of Matrix \( A \):**
\[
\text{det}(A) = \begin{vmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{vmatrix}
\]
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