The inverse of matrix `[[1, 2], [3, 4]]` is

Options

A. `(-1)/(2)[[4, -2], [-3, 1]]`
B. `(1)/(2)[[4, -2], [-3, 1]]`
C. `(-1)/(2)[[4, -3], [3, 1]]`
D. `(1)/(2)[[4, -3], [-2, 1]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \), 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 \( A \):
- \( a = 1 \)
- \( b = 2 \)
- \( c = 3 \)
- \( d = 4 \)

Calculating the determinant:
\[ \text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \]

### Step 2: Calculate the Adjoint of Matrix A
The adjoint of a 2x2 matrix is given by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. The adjoint of \( A \) is calculated as follows:
\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \]

### Step 3: Calculate the Inverse of Matrix A
The inverse of a matrix \( A \) can be calculated using the formula:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-2} \cdot \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} \]
This simplifies to:
\[ A^{-1} = \begin{bmatrix} \frac{4}{-2} & \frac{-2}{-2} \\ \frac{-3}{-2} & \frac{1}{-2} \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix} \]

### Final Answer
Thus, the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) is:
\[ A^{-1} = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix} \]