College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.5: Determinants
Problem 71E: Explain how to find the determinant of a 22 matrix.
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Question
![**Solving the Matrix Equation \(AX = B\)**
In this example, we are given the matrix equations \(A\), \(B\), and are required to solve for \(X\).
### Given Matrices:
\[ A = \begin{bmatrix} 1 & 5 \\ -2 & 8 \end{bmatrix} \]
\[ B = \begin{bmatrix} -24 \\ -24 \end{bmatrix} \]
### Matrix to Solve For:
\[ X = \begin{bmatrix} \Box \\ \Box \end{bmatrix} \]
### Steps to Solve:
1. **Identify the equation**: We have \(AX = B\).
2. **Set up the system of linear equations**:
- From the first row: \(1x_1 + 5x_2 = -24\)
- From the second row: \(-2x_1 + 8x_2 = -24\)
3. **Solve the system of linear equations**:
- We can use methods such as substitution, elimination, or matrix inversion to solve for \(x_1\) and \(x_2\).
### Solving using Matrix Inversion:
1. **Find the inverse of matrix \(A\)**:
\[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \]
- Calculate the determinant of \(A\):
\[ \text{det}(A) = (1)(8) - (5)(-2) = 8 + 10 = 18 \]
- Find the adjugate of \(A\):
\[ \text{adj}(A) = \begin{bmatrix} 8 & -5 \\ 2 & 1 \end{bmatrix} \]
- Calculate the inverse:
\[ A^{-1} = \frac{1}{18} \begin{bmatrix} 8 & -5 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} \frac{8}{18} & \frac{-5}{18} \\ \frac{2}{18} & \frac{1}{18} \end{bmatrix} \]
2. **Multiply both sides by \(A^{-1}\)** to solve for \(X\):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57708f6a-cbd4-4295-9f28-330268fec57a%2F186292d2-e777-4bc8-82fc-2811c23fb152%2F6s3wx0k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving the Matrix Equation \(AX = B\)**
In this example, we are given the matrix equations \(A\), \(B\), and are required to solve for \(X\).
### Given Matrices:
\[ A = \begin{bmatrix} 1 & 5 \\ -2 & 8 \end{bmatrix} \]
\[ B = \begin{bmatrix} -24 \\ -24 \end{bmatrix} \]
### Matrix to Solve For:
\[ X = \begin{bmatrix} \Box \\ \Box \end{bmatrix} \]
### Steps to Solve:
1. **Identify the equation**: We have \(AX = B\).
2. **Set up the system of linear equations**:
- From the first row: \(1x_1 + 5x_2 = -24\)
- From the second row: \(-2x_1 + 8x_2 = -24\)
3. **Solve the system of linear equations**:
- We can use methods such as substitution, elimination, or matrix inversion to solve for \(x_1\) and \(x_2\).
### Solving using Matrix Inversion:
1. **Find the inverse of matrix \(A\)**:
\[ A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \]
- Calculate the determinant of \(A\):
\[ \text{det}(A) = (1)(8) - (5)(-2) = 8 + 10 = 18 \]
- Find the adjugate of \(A\):
\[ \text{adj}(A) = \begin{bmatrix} 8 & -5 \\ 2 & 1 \end{bmatrix} \]
- Calculate the inverse:
\[ A^{-1} = \frac{1}{18} \begin{bmatrix} 8 & -5 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} \frac{8}{18} & \frac{-5}{18} \\ \frac{2}{18} & \frac{1}{18} \end{bmatrix} \]
2. **Multiply both sides by \(A^{-1}\)** to solve for \(X\):
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