Find a matrix X such that AX = B. A = 4 3 5 4 B = 2 -2 3-4 -2 4 Find a matrix X such that AX= B. Select the correct choice below and, if necessary, fill in the answer box to complete your cha O A. X= OB. The matrix is not invertible and therefore there is no matrix X.
Find a matrix X such that AX = B. A = 4 3 5 4 B = 2 -2 3-4 -2 4 Find a matrix X such that AX= B. Select the correct choice below and, if necessary, fill in the answer box to complete your cha O A. X= OB. The matrix is not invertible and therefore there is no matrix X.
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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![### Linear Algebra Problem: Solving for Matrix X
In this exercise, you are required to find a matrix \( X \) such that the product of matrices \( A \) and \( X \) equals matrix \( B \).
Given:
\[
A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 & -4 \\ -2 & -2 & 4 \end{bmatrix}
\]
To solve this, follow these steps:
1. **Evaluate the Invertibility of Matrix \( A \)**:
- Calculate the determinant of \( A \):
\[
\text{det}(A) = (4 \cdot 4) - (3 \cdot 5) = 16 - 15 = 1
\]
- Since the determinant is non-zero, matrix \( A \) is invertible.
2. **Compute \( A^{-1} \)**:
- Use the formula for the inverse of a 2x2 matrix:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]
- For matrix \( A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \):
\[
A^{-1} = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix}
\]
3. **Multiply \( A^{-1} \) by \( B \)**:
- Execute the matrix multiplication:
\[
X = A^{-1} B
\]
\[
X = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2 & 3 & -4 \\ -2 & -2 & 4 \end{bmatrix}
\]
4. **Calculate \( X \)**:
- Perform the multiplication to find each element of \( X \):
\[
X = \begin{bmatrix} (4 \cdot 2 + (-3) \cdot (-2))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a1a8c69-ac20-486b-9f67-8f66504c5494%2F79efaa92-db2e-4b52-a9cd-9c52b1d042cd%2Fg5zrcd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem: Solving for Matrix X
In this exercise, you are required to find a matrix \( X \) such that the product of matrices \( A \) and \( X \) equals matrix \( B \).
Given:
\[
A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 & -4 \\ -2 & -2 & 4 \end{bmatrix}
\]
To solve this, follow these steps:
1. **Evaluate the Invertibility of Matrix \( A \)**:
- Calculate the determinant of \( A \):
\[
\text{det}(A) = (4 \cdot 4) - (3 \cdot 5) = 16 - 15 = 1
\]
- Since the determinant is non-zero, matrix \( A \) is invertible.
2. **Compute \( A^{-1} \)**:
- Use the formula for the inverse of a 2x2 matrix:
\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\]
- For matrix \( A = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix} \):
\[
A^{-1} = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix}
\]
3. **Multiply \( A^{-1} \) by \( B \)**:
- Execute the matrix multiplication:
\[
X = A^{-1} B
\]
\[
X = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2 & 3 & -4 \\ -2 & -2 & 4 \end{bmatrix}
\]
4. **Calculate \( X \)**:
- Perform the multiplication to find each element of \( X \):
\[
X = \begin{bmatrix} (4 \cdot 2 + (-3) \cdot (-2))
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