Choose the correct answer from the following :
The inverse of matrix `|{:(1,3),(2,-1):}|is:`

Options

A. `1/7|{:(1,3),(2,-1):}|`
B. `1/7|{:(-1,3),(2,1):}|`
C. `1/7|{:(1,-3),(-2,-1):}|`
D. `1/7|{:(-1,-1),(2,1):}|`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 3 \\ 2 & -1 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of the Matrix
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix:
\[ \text{det}(A) = (1)(-1) - (3)(2) = -1 - 6 = -7 \]

### Step 2: Check if the Determinant is Non-Zero
Since the determinant is \(-7\), which is not equal to zero, the matrix is invertible.

### Step 3: Find the Adjoint of the Matrix
The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix:
\[ \text{adj}(A) = \begin{pmatrix} -1 & -3 \\ -2 & 1 \end{pmatrix} \]

### Step 4: Calculate the Inverse of the Matrix
The inverse of the matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-7} \cdot \begin{pmatrix} -1 & -3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{1}{7} \end{pmatrix} \]

### Final Answer
Thus, the inverse of the matrix \( \begin{pmatrix} 1 & 3 \\ 2 & -1 \end{pmatrix} \) is:
\[ A^{-1} = \begin{pmatrix} \frac{1}{7} & \frac{3}{7} \\ \frac{2}{7} & -\frac{1}{7} \end{pmatrix} \]