**Matrix Inversion Using Gauss-Jordan Method**Given Matrix:To find the inverse of matrix \( A \) using the Gauss-Jordan method, the matrix \( A \) is defined as follows:\[A = \begin{pmatrix} -3 & 0 & 0 \\ 2a & b & a \\ 10 & 0 & 2 \end{pmatrix}\]**Task:**Determine the conditions on the real numbers \( a \) and \( b \) for the inverse of \( A \), denoted as \( A^{-1} \), to exist.**Matrix Properties for Inversion:**- For a matrix to have an inverse, it must be a square matrix (which \( A \) is, as it is \( 3 \times 3 \)).- The determinant of the matrix must be non-zero. **Conclusion:**To find the conditions for the existence of \( A^{-1} \), calculate the determinant of \( A \) and ensure it is not equal to zero.

Answered: Use Gauss-Jordan to find the inverse of…

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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Question

**Matrix Inversion Using Gauss-Jordan Method**

Given Matrix:
To find the inverse of matrix \( A \) using the Gauss-Jordan method, the matrix \( A \) is defined as follows:

\[
A = \begin{pmatrix} -3 & 0 & 0 \\ 2a & b & a \\ 10 & 0 & 2 \end{pmatrix}
\]

**Task:**
Determine the conditions on the real numbers \( a \) and \( b \) for the inverse of \( A \), denoted as \( A^{-1} \), to exist.

**Matrix Properties for Inversion:**
- For a matrix to have an inverse, it must be a square matrix (which \( A \) is, as it is \( 3 \times 3 \)).
- The determinant of the matrix must be non-zero.

**Conclusion:**
To find the conditions for the existence of \( A^{-1} \), calculate the determinant of \( A \) and ensure it is not equal to zero.

Expert Solution

Step 1 Introduction

Gauss Jordan’s Matrix Inversion method.

In this method we shall find the inverse of a matrix without calculating the determinant.

In this method we shall write the augmented matrix of a square matrix by writing a unit matrix of same order as that of side by side. Then we shall transfer the matrix to a unit matrix by number of steps equal to order of the matrix and then the matrix so obtained to which the unit matrix is transferred is the inverse of the matrix .