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

Options

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

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix} \), 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 given by the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

\[ a = 1, \quad b = -1, \quad c = 2, \quad d = 3 \]

Calculating the determinant:

\[ \text{det}(A) = (1)(3) - (2)(-1) = 3 + 2 = 5 \]

### Step 2: Calculate the Adjoint of Matrix A

The adjoint of a 2x2 matrix is obtained by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. The adjoint of matrix \( A \) is given by:

\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

Substituting the values from matrix \( A \):

\[ \text{adj}(A) = \begin{bmatrix} 3 & 1 \\ -2 & 1 \end{bmatrix} \]

### Step 3: 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 determinant and the adjoint we calculated:

\[ A^{-1} = \frac{1}{5} \cdot \begin{bmatrix} 3 & 1 \\ -2 & 1 \end{bmatrix} \]

Calculating the inverse:

\[ A^{-1} = \begin{bmatrix} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{bmatrix} \]

### Final Answer

Thus, the inverse of the matrix \( A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix} \) is:

\[ A^{-1} = \begin{bmatrix} \frac{3}{5} & \frac{1}{5} \\ -\frac{2}{5} & \frac{1}{5} \end{bmatrix} \]

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