If `A=[(1,2),(3,4)]` then `A^(-1)` is equal to

Options

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

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 1 \)
- \( b = 2 \)
- \( c = 3 \)
- \( d = 4 \)

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2 \]

### Step 2: Calculate the Adjoint of 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 formula for the adjoint is:
\[ \text{Adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
\[ \text{Adj}(A) = \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]

### Step 3: Calculate the Inverse of 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 have:
\[ A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} \]
This gives:
\[ A^{-1} = \begin{pmatrix} \frac{4}{-2} & \frac{-2}{-2} \\ \frac{-3}{-2} & \frac{1}{-2} \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \]

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

---