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

Options

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

Answer

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

### Step 1: Calculate the Determinant of Matrix A

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

- \( a = 1 \)
- \( b = 2 \)
- \( c = 2 \)
- \( d = -1 \)

Calculating the determinant:

\[ \text{det}(A) = (1)(-1) - (2)(2) = -1 - 4 = -5 \]

### Step 2: Find the Adjoint of Matrix A

The adjoint of a 2x2 matrix is given by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements.

For \( A \):

\[ \text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ -2 & 1 \end{bmatrix} \]

### Step 3: Calculate the Inverse of Matrix A

The formula for the inverse of a matrix is given by:

\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \]

Substituting the values we found:

\[ A^{-1} = \frac{1}{-5} \cdot \begin{bmatrix} -1 & -2 \\ -2 & 1 \end{bmatrix} \]

### Step 4: Simplify the Inverse

Distributing \( \frac{1}{-5} \):

\[ A^{-1} = \begin{bmatrix} \frac{-1}{-5} & \frac{-2}{-5} \\ \frac{-2}{-5} & \frac{1}{-5} \end{bmatrix} = \begin{bmatrix} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \]

### Final Result

Thus, the inverse of the matrix \( A \) is:

\[ A^{-1} = \begin{bmatrix} \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix} \]

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