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 72E
Related questions
Question
![**Solving the Matrix Equation \( AX = B \) for \( X \)**
Given:
\[ A = \begin{bmatrix} 1 & 7 \\ -6 & 2 \end{bmatrix} \]
\[ B = \begin{bmatrix} 37 \\ -2 \end{bmatrix} \]
Find:
\[ X = \begin{bmatrix} \, \\ \, \end{bmatrix} \]
**Explanation:**
To solve for \( X \), we use the equation \( AX = B \). This involves calculating the inverse of matrix \( A \) (if it exists) such that \( X = A^{-1}B \). Start by finding the determinant of \( A \) and then the inverse, if applicable, followed by multiplying the inverse of \( A \) with \( B \).
**Steps:**
1. Calculate the determinant of \( A \).
2. If the determinant is non-zero, compute the inverse of \( A \).
3. Multiply the inverse of \( A \) by \( B \) to find \( X \).
Include these steps and calculations on the educational website for a comprehensive understanding of solving matrix equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05917d28-e2cf-446c-93d8-60aae139bb92%2F02f892aa-cb34-4cda-ae5e-ec484c0842c6%2F4no9vy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving the Matrix Equation \( AX = B \) for \( X \)**
Given:
\[ A = \begin{bmatrix} 1 & 7 \\ -6 & 2 \end{bmatrix} \]
\[ B = \begin{bmatrix} 37 \\ -2 \end{bmatrix} \]
Find:
\[ X = \begin{bmatrix} \, \\ \, \end{bmatrix} \]
**Explanation:**
To solve for \( X \), we use the equation \( AX = B \). This involves calculating the inverse of matrix \( A \) (if it exists) such that \( X = A^{-1}B \). Start by finding the determinant of \( A \) and then the inverse, if applicable, followed by multiplying the inverse of \( A \) with \( B \).
**Steps:**
1. Calculate the determinant of \( A \).
2. If the determinant is non-zero, compute the inverse of \( A \).
3. Multiply the inverse of \( A \) by \( B \) to find \( X \).
Include these steps and calculations on the educational website for a comprehensive understanding of solving matrix equations.
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