If `{:A=[(5,2),(3,1)]:}," then "A^(-1)=`

Options

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

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 5 & 2 \\ 3 & 1 \end{pmatrix} \), 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:
- \( a = 5 \)
- \( b = 2 \)
- \( c = 3 \)
- \( d = 1 \)

Calculating the determinant:
\[ \text{det}(A) = (5)(1) - (2)(3) = 5 - 6 = -1 \]

### Step 2: Find the Adjoint of A
The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix:
\[ \text{adj}(A) = \begin{pmatrix} 1 & -2 \\ -3 & 5 \end{pmatrix} \]

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

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

---