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

Options

A. `[[2, 1], [3, 2]]`
B. `-[[2, 1], [3, 2]]`
C. `[[2, 3], [1, 2]]`
D. `-[[2, 3], [1, 2]]`

Answer

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

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

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = (2)(2) - (-3)(-1) = 4 - 3 = 1 \]

### Step 2: Calculate the Adjoint of Matrix A
The adjoint of a 2x2 matrix is found by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal. For our matrix \( A \):
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) is given by the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the determinant and the adjoint we calculated:
\[ A^{-1} = \frac{1}{1} \cdot \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]

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

---