The inverse of matrix `[[2, 1], [1, -1]]` is

Options

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

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix} \), 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 given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 1 \)
- \( c = 1 \)
- \( d = -1 \)

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

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

### Step 3: Calculate the Inverse of Matrix 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:
\[ A^{-1} = \frac{1}{-3} \cdot \begin{bmatrix} -1 & -1 \\ -1 & 2 \end{bmatrix} \]

### Step 4: Multiply by the Scalar
Now we multiply each element of the adjoint matrix by \( \frac{1}{-3} \):
\[ A^{-1} = \begin{bmatrix} \frac{-1}{-3} & \frac{-1}{-3} \\ \frac{-1}{-3} & \frac{2}{-3} \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{2}{3} \end{bmatrix} \]

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