Let A = A 3 4] 1 2 Use t
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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Question
Find the inverse matrix
![**Matrix Inversion Tutorial**
Let \( A = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix} \). Use the formula to find its inverse.
**Matrix Inverse Calculation**
For a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the inverse \( A^{-1} \) is given by:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Plug values from matrix \( A \):
- \( a = 3 \), \( b = 4 \)
- \( c = 1 \), \( d = 2 \)
Calculate the determinant:
\[ ad - bc = 3 \cdot 2 - 4 \cdot 1 = 6 - 4 = 2 \]
Apply to inverse formula:
\[ A^{-1} = \frac{1}{2} \begin{bmatrix} 2 & -4 \\ -1 & 3 \end{bmatrix} \]
Result:
\[ A^{-1} = \begin{bmatrix} 1 & -2 \\ -0.5 & 1.5 \end{bmatrix} \]
**Complete the inverse calculation and fill in the corresponding boxes:**
- Top left: 1
- Top right: -2
- Bottom left: -0.5
- Bottom right: 1.5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e8ca5a6-77b3-4f88-a5f3-282372d2c0e0%2Fdd42e048-cec1-4c15-926f-802a4d2b0df2%2F4jgy4wf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Inversion Tutorial**
Let \( A = \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix} \). Use the formula to find its inverse.
**Matrix Inverse Calculation**
For a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the inverse \( A^{-1} \) is given by:
\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
Plug values from matrix \( A \):
- \( a = 3 \), \( b = 4 \)
- \( c = 1 \), \( d = 2 \)
Calculate the determinant:
\[ ad - bc = 3 \cdot 2 - 4 \cdot 1 = 6 - 4 = 2 \]
Apply to inverse formula:
\[ A^{-1} = \frac{1}{2} \begin{bmatrix} 2 & -4 \\ -1 & 3 \end{bmatrix} \]
Result:
\[ A^{-1} = \begin{bmatrix} 1 & -2 \\ -0.5 & 1.5 \end{bmatrix} \]
**Complete the inverse calculation and fill in the corresponding boxes:**
- Top left: 1
- Top right: -2
- Bottom left: -0.5
- Bottom right: 1.5
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