Use the adjoint method to find the inverse: 4 3 2014 5 -37 15 1 -19 15
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Understanding Matrices and Determinants
**Problem Statement:**
Given:
\[ \text{det} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = 4 \]
Find the determinants of the following matrices:
1. \[ \text{det} \begin{bmatrix} a & b & c \\ 7d & 7e & 7f \\ g & h & i \end{bmatrix} \]
2. \[ \text{det} \begin{bmatrix} a & b & c \\ d & e & f \\ g+7d & h+7e & i+7f \end{bmatrix} \]
3. \[ \text{det} \begin{bmatrix} a & b & c \\ g & h & i \\ d & e & f \end{bmatrix} \]
**Explanation:**
The initial determinant involves a 3x3 matrix with elements \(a, b, c, d, e, f, g, h, i\) and is given to be 4. Understanding properties of determinants, such as linearity and row operations, can help ease computation of these related matrices. Specifically, observe how each matrix is derived from alterations in rows which affect the determinant’s value.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2F4640e694-69b3-40ba-8c5f-b0029e7f51fe%2Fc2mw9m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Matrices and Determinants
**Problem Statement:**
Given:
\[ \text{det} \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = 4 \]
Find the determinants of the following matrices:
1. \[ \text{det} \begin{bmatrix} a & b & c \\ 7d & 7e & 7f \\ g & h & i \end{bmatrix} \]
2. \[ \text{det} \begin{bmatrix} a & b & c \\ d & e & f \\ g+7d & h+7e & i+7f \end{bmatrix} \]
3. \[ \text{det} \begin{bmatrix} a & b & c \\ g & h & i \\ d & e & f \end{bmatrix} \]
**Explanation:**
The initial determinant involves a 3x3 matrix with elements \(a, b, c, d, e, f, g, h, i\) and is given to be 4. Understanding properties of determinants, such as linearity and row operations, can help ease computation of these related matrices. Specifically, observe how each matrix is derived from alterations in rows which affect the determinant’s value.
![**Use the adjoint method to find the inverse:**
\[
\begin{pmatrix}
-1 & 3 \\
-4 & 3
\end{pmatrix}^{-1} =
\]
There is a 2x2 matrix given, and the task is to find its inverse using the adjoint method. Below the matrix, there are four options arranged in a 2x2 grid. Each option is a fraction:
Top left: \(\frac{3}{5}\)
Top right: \(-\frac{3}{5}\)
Bottom left: \(\frac{4}{15}\)
Bottom right: \(\frac{1}{15}\)
A red "X" indicates an incorrect choice is highlighted on the left of the grid.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2F4640e694-69b3-40ba-8c5f-b0029e7f51fe%2Fdrvh2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Use the adjoint method to find the inverse:**
\[
\begin{pmatrix}
-1 & 3 \\
-4 & 3
\end{pmatrix}^{-1} =
\]
There is a 2x2 matrix given, and the task is to find its inverse using the adjoint method. Below the matrix, there are four options arranged in a 2x2 grid. Each option is a fraction:
Top left: \(\frac{3}{5}\)
Top right: \(-\frac{3}{5}\)
Bottom left: \(\frac{4}{15}\)
Bottom right: \(\frac{1}{15}\)
A red "X" indicates an incorrect choice is highlighted on the left of the grid.
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