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

Options

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

Answer

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

### Step 1: Define the Matrix
Let \( A = \begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix} \).

### Step 2: Calculate the Determinant of Matrix A
The determinant of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = -1 \)
- \( b = 5 \)
- \( c = -3 \)
- \( d = 2 \)

Calculating the determinant:
\[ \text{det}(A) = (-1)(2) - (5)(-3) = -2 + 15 = 13 \]

### Step 3: Check if the Determinant is Non-Zero
Since \(\text{det}(A) = 13\) (which is not equal to zero), the inverse of matrix \( A \) exists.

### Step 4: Find the Adjoint of Matrix A
The adjoint of a \(2 \times 2\) matrix \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is given by:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]
For our matrix \( A \):
- \( d = 2 \)
- \( -b = -5 \)
- \( -c = 3 \)
- \( a = -1 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix} \]

### Step 5: Calculate the Inverse of Matrix A
The inverse of matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{13} \cdot \begin{bmatrix} 2 & -5 \\ 3 & -1 \end{bmatrix} \]
This results in:
\[ A^{-1} = \begin{bmatrix} \frac{2}{13} & -\frac{5}{13} \\ \frac{3}{13} & -\frac{1}{13} \end{bmatrix} \]

### Final Answer
The inverse of the matrix \(\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}\) is:
\[ A^{-1} = \begin{bmatrix} \frac{2}{13} & -\frac{5}{13} \\ \frac{3}{13} & -\frac{1}{13} \end{bmatrix} \]

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