If `A=[(5,4),(3,2)]` then `A^(-1)` is equal to

Options

A. `(1)/(2)[(-2,4),(3,5)]`
B. `(1)/(2)[(5,3),(4,2)]`
C. `(1)/(2)[(5,-4),(-3,2)]`
D. `-(1)/(2)[(2,-4),(-3,5)]`

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 5 & 4 \\ 3 & 2 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The formula for the determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 5 \)
- \( b = 4 \)
- \( c = 3 \)
- \( d = 2 \)

Calculating the determinant:
\[ \text{det}(A) = (5)(2) - (4)(3) = 10 - 12 = -2 \]

### Step 2: Check if the Inverse Exists
An inverse exists if the determinant is not equal to zero. Since \( \text{det}(A) = -2 \neq 0 \), the inverse exists.

### Step 3: Calculate the Adjoint of Matrix A
For a 2x2 matrix, the adjoint is found by swapping the elements on the main diagonal and changing the signs of the elements on the other diagonal. Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 2 & -4 \\ -3 & 5 \end{pmatrix} \]

### Step 4: Calculate the Inverse of Matrix A
The formula for the inverse of a matrix \( A \) is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]
Substituting the values we found:
\[ A^{-1} = \frac{1}{-2} \cdot \begin{pmatrix} 2 & -4 \\ -3 & 5 \end{pmatrix} = \begin{pmatrix} \frac{2}{-2} & \frac{-4}{-2} \\ \frac{-3}{-2} & \frac{5}{-2} \end{pmatrix} = \begin{pmatrix} -1 & 2 \\ \frac{3}{2} & -\frac{5}{2} \end{pmatrix} \]

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

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