In Exercises 11-12, use the inversion algorithm to find the inverse of the matrix (if the inverse exists). 1 2 3 2 5 3 1 08. 11. a. b. -1 2 -4 3 -4 4 1 2 -9
In Exercises 11-12, use the inversion algorithm to find the inverse of the matrix (if the inverse exists). 1 2 3 2 5 3 1 08. 11. a. b. -1 2 -4 3 -4 4 1 2 -9
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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![### Matrix Inversion Exercises
#### Instructions:
In Exercises **11–12**, use the inversion algorithm to find the inverse of the matrix (if the inverse exists).
#### Exercise 11
**a.** Find the inverse of the following matrix:
\[
\begin{pmatrix}
1 & 2 & 3 \\
2 & 5 & 3 \\
1 & 0 & 8
\end{pmatrix}
\]
**b.** Find the inverse of the following matrix:
\[
\begin{pmatrix}
-1 & 3 & -4 \\
2 & 4 & 1 \\
-4 & 2 & -9
\end{pmatrix}
\]
### Steps to Calculate the Inverse of a Matrix (If it Exists):
1. **Form the augmented matrix:** Combine the given matrix with the identity matrix of the same dimension.
2. **Row reduce the augmented matrix:** Use row operations to transform the left part of the augmented matrix (the given matrix) into the identity matrix.
3. **Result:** The right part of the augmented matrix will be the inverse of the original matrix if the transformations are possible.
Remember, a matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcabf4e2-d726-470f-ab0b-ceee8041df2c%2F1c765488-e697-4114-ae55-bbf3267784df%2Fx2k47k_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Inversion Exercises
#### Instructions:
In Exercises **11–12**, use the inversion algorithm to find the inverse of the matrix (if the inverse exists).
#### Exercise 11
**a.** Find the inverse of the following matrix:
\[
\begin{pmatrix}
1 & 2 & 3 \\
2 & 5 & 3 \\
1 & 0 & 8
\end{pmatrix}
\]
**b.** Find the inverse of the following matrix:
\[
\begin{pmatrix}
-1 & 3 & -4 \\
2 & 4 & 1 \\
-4 & 2 & -9
\end{pmatrix}
\]
### Steps to Calculate the Inverse of a Matrix (If it Exists):
1. **Form the augmented matrix:** Combine the given matrix with the identity matrix of the same dimension.
2. **Row reduce the augmented matrix:** Use row operations to transform the left part of the augmented matrix (the given matrix) into the identity matrix.
3. **Result:** The right part of the augmented matrix will be the inverse of the original matrix if the transformations are possible.
Remember, a matrix must be square (same number of rows and columns) and have a non-zero determinant to have an inverse.
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