`[{:(2,1),(1,1):}]` find the inverse of matrix

Explanation

To find the inverse of the given matrix \( A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \), 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 calculated using the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix:
\[ \text{det}(A) = (2)(1) - (1)(1) = 2 - 1 = 1 \]

### Step 2: Calculate 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 our matrix:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} \]

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

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

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