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is there a mistake here orginal matrix was 2x2 (2,3 1,2) finding elementary matrixs

Let's review the Gauss-Jordan elimination process shown in the image to determine if any mistakes were made while finding the elementary matrices and their inverses for the original \( 2 \times 2 \) matrix \( A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \).

### Steps in the image:

1. **Original matrix**: 
   \( A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \)

2. **First step** (row swap \( R_1 \leftrightarrow R_2 \)):
   This transforms \( A \) into:
   \[
   \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}
   \]
   The elementary matrix for this step is:
   \[
   E_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
   \]

3. **Second step** (row operation \( R_2 \rightarrow R_2 - 2R_1 \)):
   This transforms the matrix into:
   \[
   \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}
   \]
   The elementary matrix for this step is:
   \[
   E_2 = \begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix}
   \]

4. **Third step** (row operation \( R_2 \rightarrow -R_2 \)):
   This transforms the matrix into:
   \[
   \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}
   \]
   The elementary matrix for this step is:
   \[
   E_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
   \]

5. **Fourth step** (row operation \( R_1 \rightarrow R_1 - 2R_2 \)):
   This gives the identity matrix:
   \[
   \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
   \]
   The elementary matrix for this step is:
   \[
   E_4 = \begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix}
   \]

### Observations:
The elementary matrices seem correct, and their inverses are computed correctly as well.

### Answer:
It looks like there is **no mistake** in the Gauss-Jordan elimination process in this image. The steps and elementary matrices match the standard operations. Each matrix manipulation is consistent with the expected transformation steps.

Would you like more details or have any further questions?

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Here are 5 related questions to deepen your understanding:
1. How can you verify the inverse of an elementary matrix?
2. Why is row swapping an important first step in Gauss-Jordan elimination?
3. Can the inverse of a matrix be found without using elementary matrices?
4. How would you perform Gauss-Jordan elimination on a larger matrix?
5. What happens if the matrix is singular (non-invertible)?

**Tip**: When working with matrices, always double-check each row operation to ensure accuracy in transformations and elementary matrix formation.

Is there a mistake here? Original matrix was 2x2 (2,3 1,2) finding elementary matrices.

Explore the step-by-step process of Gauss-Jordan elimination to find elementary matrices for the 2x2 matrix (2,3 1,2). Review the row operations involved and check for mistakes in the calculation. Learn about elementary matrices and their inverses in this detailed solution.

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