If `A=[(3,1),(-1,2)]`, find `A^(-1)`

Options

A. `1/7[(2,-1),(1,3)]`
B. `1/7[(3,1),(-1,2)]`
C. `1/7[(-2,1),(-1,-3)]`
D. `1/7[(2,1),(1,-3)]`

Answer

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

### Step 1: Calculate the Determinant of Matrix 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 \( A \):
- \( a = 3 \)
- \( b = 1 \)
- \( c = -1 \)
- \( d = 2 \)

Now, we can calculate the determinant:
\[ \text{det}(A) = (3)(2) - (1)(-1) = 6 + 1 = 7 \]

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

### 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:
\[ A^{-1} = \frac{1}{7} \cdot \begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix} \]

### Step 4: Write the Final Result
Thus, the inverse of matrix \( A \) is:
\[ A^{-1} = \begin{pmatrix} \frac{2}{7} & \frac{-1}{7} \\ \frac{1}{7} & \frac{3}{7} \end{pmatrix} \]