The inverse of the matrix `[ (5,-2),(3,1)]` is

Options

A. `(1)/(11) [(1,2),(-3,5)]`
B. `[(1,2),(-3,5)]`
C. `(1)/(13)[(-2,5),(1,3)]`
D. `[(1,3),(-2,5)]`

Answer

To find the inverse of the matrix \(\begin{pmatrix} 5 & -2 \\ 3 & 1 \end{pmatrix}\), we will follow these steps:

### Step 1: Identify the elements of the matrix
Let the matrix \(A\) be:
\[ A = \begin{pmatrix} 5 & -2 \\ 3 & 1 \end{pmatrix} \]
Here, we have:
- \(a = 5\)
- \(b = -2\)
- \(c = 3\)
- \(d = 1\)

### Step 2: Calculate the determinant of the matrix
The determinant of a \(2 \times 2\) matrix is given by the formula:
\[ \text{det}(A) = ad - bc \]
Substituting the values:
\[ \text{det}(A) = (5)(1) - (-2)(3) = 5 + 6 = 11 \]

### Step 3: Use the formula for the inverse of a \(2 \times 2\) matrix
The inverse of a \(2 \times 2\) matrix is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
Substituting the values of \(a\), \(b\), \(c\), and \(d\):
\[ A^{-1} = \frac{1}{11} \begin{pmatrix} 1 & 2 \\ -3 & 5 \end{pmatrix} \]

### Step 4: Write the final answer
Thus, the inverse of the matrix \(\begin{pmatrix} 5 & -2 \\ 3 & 1 \end{pmatrix}\) is:
\[ A^{-1} = \begin{pmatrix} \frac{1}{11} & \frac{2}{11} \\ -\frac{3}{11} & \frac{5}{11} \end{pmatrix} \]