Need question 5 ### Linear Algebra: Matrix Exercises#### Exercise 11. Find the inverse of the matrix: \[ \begin{bmatrix} 3 & 2 \\ 8 & 5 \end{bmatrix} \]#### Exercise 22. For what values of $ h $ is the matrix: \[ \begin{bmatrix} 1 & h \\ -1 & h \end{bmatrix} \] invertible?#### Exercise 33. Suppose that $ P $ is invertible and that $ A = PBP^{-1} $. Solve for $ B $ in terms of $ A $ and $ P $.#### Exercise 44. Suppose that $ AB = AC $, where $ B $ and $ C $ are $ n \times p $ matrices. Is $ B = C $? What if we also assume that $ A $ is invertible?---#### Exercise 55. Find the inverse of the matrix: \[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} \]

Consider given matrix A=1000110011101111. Write the matrix in augmented form.…

Answered: 5. Find the inverse of 100 07 1 10 0 1…

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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Need question 5

### Linear Algebra: Matrix Exercises

#### Exercise 1
1. Find the inverse of the matrix:
\[
\begin{bmatrix}
3 & 2 \\
8 & 5
\end{bmatrix}
\]

#### Exercise 2
2. For what values of $ h $ is the matrix:
\[
\begin{bmatrix}
1 & h \\
-1 & h
\end{bmatrix}
\]
invertible?

#### Exercise 3
3. Suppose that $ P $ is invertible and that $ A = PBP^{-1} $. Solve for $ B $ in terms of $ A $ and $ P $.

#### Exercise 4
4. Suppose that $ AB = AC $, where $ B $ and $ C $ are $ n \times p $ matrices. Is $ B = C $? What if we also assume that $ A $ is invertible?

---

#### Exercise 5
5. Find the inverse of the matrix:
\[
\begin{bmatrix}
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 1
\end{bmatrix}
\]

Expert Solution

Step 1

Consider given matrix $A = [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix}]$ .

Write the matrix in augmented form.

$[\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix} \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$

Perform operation $R_{2} \to R_{2} - R_{1}$ .

$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix} \begin{matrix} 1 & 0 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$

Perform operation $R_{3} \to R_{3} - R_{1}$ .

$[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{matrix} \begin{matrix} 1 & 0 & 0 & 0 \\ - 1 & 1 & 0 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$