find the inverse of matrix `A=[{:(4,7),(3,5):}]` by using elementary column transformation .

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 4 & 7 \\ 3 & 5 \end{pmatrix} \) using elementary column transformations, we will follow these steps:

### Step 1: Set up the augmented matrix
We will set up the augmented matrix \([A | I]\), where \(I\) is the identity matrix of the same size as \(A\).

\[ \left( \begin{array}{cc|cc} 4 & 7 & 1 & 0 \\ 3 & 5 & 0 & 1 \end{array} \right) \]

### Step 2: Make the first column's leading entry 1
To make the leading entry of the first column equal to 1, we can divide the first column by 4.

\[ C_1 \leftarrow \frac{1}{4} C_1 \]

This gives us:

\[ \left( \begin{array}{cc|cc} 1 & 7/4 & 1/4 & 0 \\ 3 & 5 & 0 & 1 \end{array} \right) \]

### Step 3: Eliminate the entry below the leading 1
Next, we need to eliminate the entry below the leading 1 in the first column. We can do this by replacing the second row with \( R_2 - 3R_1 \).

\[ R_2 \leftarrow R_2 - 3R_1 \]

Calculating this gives:

\[ \left( \begin{array}{cc|cc} 1 & 7/4 & 1/4 & 0 \\ 0 & -1/4 & -3/4 & 1 \end{array} \right) \]

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

\[ R_2 \leftarrow -4R_2 \]

This results in:

\[ \left( \begin{array}{cc|cc} 1 & 7/4 & 1/4 & 0 \\ 0 & 1 & 3 & -4 \end{array} \right) \]

### Step 5: Eliminate the entry above the leading 1
Now we need to eliminate the entry above the leading 1 in the second column. We can do this by replacing the first row with \( R_1 - \frac{7}{4} R_2 \).

\[ R_1 \leftarrow R_1 - \frac{7}{4} R_2 \]

Calculating this gives:

\[ \left( \begin{array}{cc|cc} 1 & 0 & -5 & \frac{7}{4} \\ 0 & 1 & 3 & -4 \end{array} \right) \]

### Step 6: Write the inverse matrix
Now that we have transformed the left side into the identity matrix, the right side gives us the inverse of matrix \( A \):

\[ A^{-1} = \begin{pmatrix} -5 & \frac{7}{4} \\ 3 & -4 \end{pmatrix} \]

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

\[ A^{-1} = \begin{pmatrix} -5 & \frac{7}{4} \\ 3 & -4 \end{pmatrix} \]