**Exercise 13**\[ \begin{bmatrix}3 & 5 & 4 \\1 & 0 & 1 \\2 & 1 & 1 \end{bmatrix} \]---In Exercises 11–16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix. ## Theorem 8: An Inverse FormulaLet $ A $ be an invertible $ n \times n $ matrix. Then\[ A^{-1} = \frac{1}{\det A} \text{adj } A \]Where:- $ A^{-1} $ is the inverse of matrix $ A $.- $ \det A $ is the determinant of matrix $ A $.- $ \text{adj } A $ is the adjugate (or adjoint) of matrix $ A $.This theorem provides a formula to compute the inverse of an invertible matrix using its determinant and its adjugate.

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3 13. In Exercises 11-16,compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

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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Question

**Exercise 13**

\[
\begin{bmatrix}
3 & 5 & 4 \\
1 & 0 & 1 \\
2 & 1 & 1
\end{bmatrix}
\]

---

In Exercises 11–16, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

## Theorem 8: An Inverse Formula

Let $ A $ be an invertible $ n \times n $ matrix. Then

\[ A^{-1} = \frac{1}{\det A} \text{adj } A \]

Where:
- $ A^{-1} $ is the inverse of matrix $ A $.
- $ \det A $ is the determinant of matrix $ A $.
- $ \text{adj } A $ is the adjugate (or adjoint) of matrix $ A $.

This theorem provides a formula to compute the inverse of an invertible matrix using its determinant and its adjugate.

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