**Title: Verifying the Inverse of a Matrix****Objective:** Demonstrate that matrix $ B $ is the inverse of matrix $ A $.---**Given Matrices:**Matrix $ A $ is:\[A = \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix}\]Matrix $ B $ is proposed as the inverse of $ A $:\[B = \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix}\]---**Verification Process:**To verify $ B $ is the inverse of $ A $, we need to show:1. $ AB = I $2. $ BA = I $Where $ I $ is the identity matrix:\[I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]---**Calculations:**1. **Calculate $ AB $:** \[ AB = \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix} \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix} = \begin{bmatrix} \text{(To be filled)} & \text{(To be filled)} \\ \text{(To be filled)} & \text{(To be filled)} \end{bmatrix} = I \]2. **Calculate $ BA $:** \[ BA = \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix} = \begin{bmatrix} \text{(To be filled)} & \text{(To be filled)} \\ \text{(To be filled)} & \text{(To be filled)} \end{bmatrix} = I \]---**Diagram Explanation:**The image contains empty boxes to be filled during the matrix multiplication process. Green arrows indicate the intended calculations resulting in the identity matrix.

Given data: The first matrix given is A=1258. The second matrix given is B=-415/2-1/2.…

Answered: Show that B is the inverse of A. A [5 8

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Title: Verifying the Inverse of a Matrix**

**Objective:** Demonstrate that matrix $ B $ is the inverse of matrix $ A $.

---

**Given Matrices:**

Matrix $ A $ is:

\[
A = \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix}
\]

Matrix $ B $ is proposed as the inverse of $ A $:

\[
B = \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix}
\]

---

**Verification Process:**

To verify $ B $ is the inverse of $ A $, we need to show:

1. $ AB = I $
2. $ BA = I $

Where $ I $ is the identity matrix:

\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

---

**Calculations:**

1. **Calculate $ AB $:**

\[
AB = \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix} \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix} = \begin{bmatrix} \text{(To be filled)} & \text{(To be filled)} \\ \text{(To be filled)} & \text{(To be filled)} \end{bmatrix} = I
\]

2. **Calculate $ BA $:**

\[
BA = \begin{bmatrix} -4 & 1 \\ \frac{5}{2} & -\frac{1}{2} \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 5 & 8 \end{bmatrix} = \begin{bmatrix} \text{(To be filled)} & \text{(To be filled)} \\ \text{(To be filled)} & \text{(To be filled)} \end{bmatrix} = I
\]

---

**Diagram Explanation:**

The image contains empty boxes to be filled during the matrix multiplication process. Green arrows indicate the intended calculations resulting in the identity matrix.

Expert Solution

Step 1

Given data:

The first matrix given is $A = [\begin{matrix} 1 & 2 \\ 5 & 8 \end{matrix}]$ .

The second matrix given is $B = [\begin{matrix} - 4 & 1 \\ 5 / 2 & - 1 / 2 \end{matrix}]$ .

Evaluate the value of the AB matrix.

$\begin{array}{rcl} A B & = & [\begin{matrix} 1 & 2 \\ 5 & 8 \end{matrix}] [\begin{matrix} - 4 & 1 \\ 5 / 2 & - 1 / 2 \end{matrix}] \\ = & [\begin{matrix} (- 4 + 2 (5 / 2)) & (1 + 2 (- 1 / 2)) \\ (5 (- 4) + 8 (5 / 2)) & (5 + 8 (- 1 / 2)) \end{matrix}] \\ = & [\begin{matrix} (- 4 + 5) & (1 - 1) \\ (- 20 + 20) & (5 - 4) \end{matrix}] \\ = & [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{array}$

This is equal to the identity matrix of order 2.

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