https://file.aiquickdraw.com/mathbot/file/1727074811031039q6gxe.jpg

The problem is asking to find the inverse of the matrix:

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

using the **Gauss-Jordan Method**. 

### Steps for using the Gauss-Jordan method:
1. Write the given matrix \( A \) and append the identity matrix \( I \) of the same order to it, forming an augmented matrix \( [A | I] \).
2. Perform row operations (using elementary row operations) to transform the matrix \( A \) into the identity matrix \( I \).
3. The matrix that replaces \( I \) on the right will be the inverse matrix \( A^{-1} \).

#### Step 1: Set up the augmented matrix

\[
\left[ \begin{array}{ccc|ccc}
2 & 0 & 3 & 1 & 0 & 0 \\
-3 & -1 & -3 & 0 & 1 & 0 \\
4 & 1 & 4 & 0 & 0 & 1
\end{array} \right]
\]

#### Step 2: Perform row operations
We will now perform row operations to make the left matrix into the identity matrix. I will calculate this step-by-step.

Let me do the computation.The inverse of the given matrix using the Gauss-Jordan method is:

\[
A^{-1} = \begin{bmatrix}
-1 & 3 & 3 \\
0 & -4 & -3 \\
1 & -2 & -2
\end{bmatrix}
\]

Would you like further details on how any of the steps were done, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How do elementary row operations work in the Gauss-Jordan method?
2. What are the conditions for a matrix to be invertible?
3. Can you use the Gauss-Jordan method to solve a system of linear equations?
4. How does the Gauss-Jordan method differ from Gaussian elimination?
5. What is the significance of the identity matrix in matrix inversion?

**Tip:** When dealing with matrices, always verify that the determinant is non-zero to ensure that the matrix is invertible!

Using Gauss Jordan Method, find the inverse of the matrix [ [2, 0, 3], [-3, -1, -3], [4, 1, 4] ]

This page explains how to use the Gauss-Jordan method to find the inverse of a 3x3 matrix, specifically [ [2, 0, 3], [-3, -1, -3], [4, 1, 4] ]. Learn the step-by-step process, including setting up an augmented matrix and performing elementary row operations.

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