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find inverse of the given matix a in gauss jordan method

The given matrix \( A \) is:

\[
A = \begin{bmatrix} 0 & 3 & -4 \\ 5 & 6 & 7 \\ 2 & 0 & 8 \end{bmatrix}
\]

To find the inverse of the matrix \( A \) using the Gauss-Jordan method, we will perform row operations to transform \( A \) into the identity matrix, while simultaneously applying these operations to the identity matrix to obtain \( A^{-1} \).

### Steps to Find the Inverse Using the Gauss-Jordan Method:

1. **Augment the matrix \( A \) with the identity matrix**:

\[
\left[\begin{array}{ccc|ccc}
0 & 3 & -4 & 1 & 0 & 0 \\
5 & 6 & 7 & 0 & 1 & 0 \\
2 & 0 & 8 & 0 & 0 & 1
\end{array}\right]
\]

2. **Use row operations to make the augmented matrix into reduced row echelon form**:

   - First, swap Row 1 and Row 2 to bring a non-zero element to the top left:
   
   \[
   \left[\begin{array}{ccc|ccc}
   5 & 6 & 7 & 0 & 1 & 0 \\
   0 & 3 & -4 & 1 & 0 & 0 \\
   2 & 0 & 8 & 0 & 0 & 1
   \end{array}\right]
   \]

   - Normalize Row 1 by dividing by 5:

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\
   0 & 3 & -4 & 1 & 0 & 0 \\
   2 & 0 & 8 & 0 & 0 & 1
   \end{array}\right]
   \]

   - Eliminate the first column below Row 1:

     - For Row 3: \( R3 = R3 - 2 \times R1 \)

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\
   0 & 3 & -4 & 1 & 0 & 0 \\
   0 & -2.4 & 5.2 & 0 & -\frac{2}{5} & 1
   \end{array}\right]
   \]

   - Next, normalize Row 2 by dividing by 3:

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\
   0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\
   0 & -2.4 & 5.2 & 0 & -\frac{2}{5} & 1
   \end{array}\right]
   \]

   - Eliminate Row 2's contribution from Row 1 and Row 3:

     - For Row 1: \( R1 = R1 - 1.2 \times R2 \)
     - For Row 3: \( R3 = R3 + 2.4 \times R2 \)

   After the above operations:

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 0 & 3 & -0.4 & \frac{1}{5} & 0 \\
   0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\
   0 & 0 & 2 & \frac{4}{5} & -\frac{2}{5} & 1
   \end{array}\right]
   \]

   - Finally, normalize the last row and adjust the first two rows:

     - Normalize Row 3 by dividing by 2:

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 0 & 3 & -0.4 & \frac{1}{5} & 0 \\
   0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\
   0 & 0 & 1 & \frac{2}{5} & -\frac{1}{5} & \frac{1}{2}
   \end{array}\right]
   \]

     - Adjust Rows 1 and 2 to eliminate the third column:

     - For Row 1: \( R1 = R1 - 3 \times R3 \)
     - For Row 2: \( R2 = R2 + \frac{4}{3} \times R3 \)

   \[
   \left[\begin{array}{ccc|ccc}
   1 & 0 & 0 & -2 & 1 & -\frac{3}{2} \\
   0 & 1 & 0 & 1 & -\frac{1}{2} & 1 \\
   0 & 0 & 1 & \frac{2}{5} & -\frac{1}{5} & \frac{1}{2}
   \end{array}\right]
   \]

Thus, the inverse of matrix \( A \) is:

\[
A^{-1} = \begin{bmatrix} -2 & 1 & -\frac{3}{2} \\ 1 & -\frac{1}{2} & 1 \\ \frac{2}{5} & -\frac{1}{5} & \frac{1}{2} \end{bmatrix}
\]

Do you have any questions or need further details?

### Additional Related Questions:

1. What are the conditions for a matrix to have an inverse?
2. How can you verify that a matrix inverse is correct?
3. What are other methods to find the inverse of a matrix apart from the Gauss-Jordan method?
4. What does it mean if a matrix does not have an inverse?
5. How does the determinant of a matrix relate to the existence of its inverse?

**Tip:** Always check the determinant of the matrix first; if it's zero, the matrix does not have an inverse.

Find the inverse of the given matrix A using the Gauss-Jordan method.

This detailed solution explains how to find the inverse of a 3x3 matrix using the Gauss-Jordan elimination method. Key steps include augmenting the matrix with an identity matrix, performing row operations, and transforming the matrix into reduced row echelon form. Suitable for high school and college students.

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