`[{:(2,3),(5,7):}]` find the inverse of the matrix

Explanation

To find the inverse of the matrix \(\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix}\), we will use the method of row operations. Let's denote the matrix as \(A\).

### Step 1: Set up the augmented matrix
We start with the matrix \(A\) and augment it with the identity matrix:

\[ \left( \begin{array}{cc|cc} 2 & 3 & 1 & 0 \\ 5 & 7 & 0 & 1 \end{array} \right) \]

### Step 2: Make the leading coefficient of the first row equal to 1
To make the leading coefficient of the first row equal to 1, we divide the entire first row by 2:

\[ R_1 \rightarrow \frac{1}{2} R_1 \]

This gives us:

\[ \left( \begin{array}{cc|cc} 1 & \frac{3}{2} & \frac{1}{2} & 0 \\ 5 & 7 & 0 & 1 \end{array} \right) \]

### Step 3: Eliminate the first element of the second row
Next, we want to eliminate the first element of the second row. We can do this by replacing the second row with \(R_2 - 5R_1\):

\[ R_2 \rightarrow R_2 - 5R_1 \]

Calculating this gives:

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

### Step 4: Make the leading coefficient of the second row equal to 1
Now, we need to make the leading coefficient of the second row equal to 1. We can do this by multiplying the second row by -2:

\[ R_2 \rightarrow -2R_2 \]

This results in:

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

### Step 5: Eliminate the second element of the first row
Next, we want to eliminate the second element of the first row. We can do this by replacing the first row with \(R_1 - \frac{3}{2}R_2\):

\[ R_1 \rightarrow R_1 - \frac{3}{2}R_2 \]

Calculating this gives:

\[ \left( \begin{array}{cc|cc} 1 & 0 & -\frac{7}{2} & 3 \\ 0 & 1 & 5 & -2 \end{array} \right) \]

### Step 6: Write the inverse matrix
Now we can write the inverse matrix from the augmented part:

\[ A^{-1} = \begin{pmatrix} -\frac{7}{2} & 3 \\ 5 & -2 \end{pmatrix} \]

### Final Answer
The inverse of the matrix \(\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix}\) is:

\[ A^{-1} = \begin{pmatrix} -\frac{7}{2} & 3 \\ 5 & -2 \end{pmatrix} \]