Solve the matrix equation AX=B for X. 1 5 -24 A = B = -2 8 -24

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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**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 \(X\):
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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