`[{:(2,5),(1,3):}]` using elementary method find the inverse of the matrix

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \) using the elementary method, we will augment the matrix \( A \) with the identity matrix and perform row operations to transform \( A \) into the identity matrix. The augmented matrix will be:

\[ \begin{pmatrix} 2 & 5 & | & 1 & 0 \\ 1 & 3 & | & 0 & 1 \end{pmatrix} \]

### Step 1: Make the leading coefficient of the first row equal to 1.

To do this, we can divide the second row by 1 (which doesn't change it) and then use it to eliminate the first element of the first row.

\[ R_1 = R_1 - 2R_2 \]

Calculating \( R_1 \):

- First element: \( 2 - 2 \cdot 1 = 0 \)
- Second element: \( 5 - 2 \cdot 3 = -1 \)

So, the new augmented matrix becomes:

\[ \begin{pmatrix} 0 & -1 & | & 1 & -2 \\ 1 & 3 & | & 0 & 1 \end{pmatrix} \]

### Step 2: Swap the rows to have a leading 1 in the first row.

Now we swap \( R_1 \) and \( R_2 \):

\[ \begin{pmatrix} 1 & 3 & | & 0 & 1 \\ 0 & -1 & | & 1 & -2 \end{pmatrix} \]

### Step 3: Make the leading coefficient of the second row equal to 1.

To do this, we can multiply the second row by -1:

\[ R_2 = -R_2 \]

Calculating \( R_2 \):

- First element: \( 0 \)
- Second element: \( 1 \)
- Third element: \( -1 \)
- Fourth element: \( 2 \)

So, the new augmented matrix becomes:

\[ \begin{pmatrix} 1 & 3 & | & 0 & 1 \\ 0 & 1 & | & -1 & 2 \end{pmatrix} \]

### Step 4: Eliminate the second element in the first row.

Now we will eliminate the second element in the first row by performing:

\[ R_1 = R_1 - 3R_2 \]

Calculating \( R_1 \):

- First element: \( 1 - 3 \cdot 0 = 1 \)
- Second element: \( 3 - 3 \cdot 1 = 0 \)
- Third element: \( 0 - 3 \cdot (-1) = 3 \)
- Fourth element: \( 1 - 3 \cdot 2 = -5 \)

So, the new augmented matrix becomes:

\[ \begin{pmatrix} 1 & 0 & | & 3 & -5 \\ 0 & 1 & | & -1 & 2 \end{pmatrix} \]

### Final Step: Write the inverse matrix.

The left side of the augmented matrix is now the identity matrix, and the right side gives us the inverse of \( A \):

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

### Summary of Steps:
1. Augment the matrix with the identity matrix.
2. Use row operations to convert the left side to the identity matrix.
3. The right side of the augmented matrix will be the inverse.