To find the inverse of the matrix $\mathbf{A} = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$ using the Gauss-Jordan method, we augment $\mathbf{A}$ with the identity matrix of the same order and perform row operations until the left side becomes the identity matrix. The right side will then be the inverse of $\mathbf{A}$.

The steps are as follows:

1. Write the augmented matrix:

   2 & 0 & -1 & 1 & 0 & 0 \\ 5 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \\ \end{array}\right]$$

2. Make the first element of the first column (pivot) equal to 1 by dividing the first row by 2:

   1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 5 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \\ \end{array}\right]$$

3. Eliminate the entries below the pivot in the first column by performing row operations:

   • $R_2 = R_2 - 5 \times R_1$
   • $R_3$ is already 0 in the first column, so no change is needed.

   This gives:

   1 & 0 & -\frac{1}{2} & \frac{1}{2} & 0 & 0 \\ 0 & 1 & \frac{5}{2} & -\frac{5}{2} & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 & 1 \\ \end{array}\right]$$

4. Make the pivot in the second row (second column) equal to 1: $R_2 = R_2 - R_3$