Can you explain how to get these answers please ? **Matrix Inverse Calculations**Consider the matrix $ A = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} $, which has the inverse $ \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} $. Determine the following. *Hint: Use the algebraic properties of matrix inverse.***(a) $(3A)^{-1}$:**The inverse of the scaled matrix $ 3A $ is:\[\begin{bmatrix} 2/3 & -5/3 \\ -1/3 & 1 \end{bmatrix}\]**(b) $(A^2)^{-1}$:**The inverse of the squared matrix $ A^2 $ is:\[\begin{bmatrix} 9 & -25 \\ -5 & 14 \end{bmatrix}\]**(c) $A^{-2}$:**No values provided in the matrix.**(d) $(A^{5})^{-1}$:**The inverse of the fifth power of matrix $ A $ is:\[\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}\]

The algebraic property of the matrix inverse are (kA)-1 = 1kA-1 , where k is the constant. An-1 =…

Answered: [} 3 5 2 -5 Consider the matrix A =…

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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Can you explain how to get these answers please ?

**Matrix Inverse Calculations**

Consider the matrix $ A = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} $, which has the inverse $ \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} $. Determine the following. *Hint: Use the algebraic properties of matrix inverse.*

**(a) $(3A)^{-1}$:**

The inverse of the scaled matrix $ 3A $ is:

\[
\begin{bmatrix} 2/3 & -5/3 \\ -1/3 & 1 \end{bmatrix}
\]

**(b) $(A^2)^{-1}$:**

The inverse of the squared matrix $ A^2 $ is:

\[
\begin{bmatrix} 9 & -25 \\ -5 & 14 \end{bmatrix}
\]

**(c) $A^{-2}$:**

No values provided in the matrix.

**(d) $(A^{5})^{-1}$:**

The inverse of the fifth power of matrix $ A $ is:

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

Expert Solution

Step 1

The algebraic property of the matrix inverse are

${(k A)}^{- 1} = \frac{1}{k} A^{- 1}$ , where $k$ is the constant.
${(A^{n})}^{- 1} = {(A^{- 1})}^{n}$
$A^{- n} = {(A^{- 1})}^{n}$
${(A^{t})}^{- 1} = {(A^{- 1})}^{t}$

(a)

$\begin{array}{rcl} {(3 A)}^{- 1} & = & \frac{1}{3} A^{- 1} \\ = & \frac{1}{3} [\begin{matrix} 2 & - 5 \\ - 1 & 3 \end{matrix}] \\ = & [\begin{matrix} \frac{2}{3} & \frac{- 5}{3} \\ \frac{- 1}{3} & 1 \end{matrix}] \end{array}$

(b)

$\begin{array}{rcl} {(A^{2})}^{- 1} & = & {(A^{- 1})}^{2} \\ = & (A^{- 1}) (A^{- 1}) \\ = & [\begin{matrix} 2 & - 5 \\ - 1 & 3 \end{matrix}] [\begin{matrix} 2 & - 5 \\ - 1 & 3 \end{matrix}] \\ = & [\begin{matrix} (2) (2) + (- 5) (- 1) & (2) (- 5) + (- 5) (3) \\ (- 1) (2) + (3) (- 1) & (- 1) (- 5) + (3) (3) \end{matrix}] \\ = & [\begin{matrix} 4 + 5 & - 10 - 15 \\ - 2 - 3 & 5 + 9 \end{matrix}] \\ = & [\begin{matrix} 9 & - 25 \\ - 5 & 14 \end{matrix}] \end{array}$