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

Options

A. `(-1)/(2)[[3, 1, -1], [1, 3, 1], [-1, 1, 3]]`
B. `(1)/(2)[[3, 1, -1], [1, 3, 1], [-1, 1, 3]]`
C. `(-1)/(4)[[3, 1, -1], [1, 3, 1], [-1, 1, 3]]`
D. `(1)/(4)[[3, 1, -1], [1, 3, 1], [-1, 1, 3]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix} \), we will follow the steps of calculating the determinant and the adjoint of the matrix, and then use these to find the inverse.

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

The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) can be calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

\[ \text{det}(A) = 2 \cdot (2 \cdot 2 - (-1) \cdot (-1)) - (-1) \cdot (-1 \cdot 2 - (-1) \cdot 1) + 1 \cdot (-1 \cdot (-1) - 2 \cdot 1) \]

Calculating each term:

1. \( 2 \cdot (4 - 1) = 2 \cdot 3 = 6 \)
2. \( -(-1) \cdot (-2 + 1) = 1 \cdot (-1) = -1 \)
3. \( 1 \cdot (1 - 2) = 1 \cdot (-1) = -1 \)

Putting it all together:

\[ \text{det}(A) = 6 - 1 - 1 = 4 \]

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

The adjoint of a matrix is the transpose of its cofactor matrix. We will calculate the cofactor for each element of the matrix.

#### Cofactor Calculation

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = \text{det}\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} = (2 \cdot 2) - (-1 \cdot -1) = 4 - 1 = 3 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = -\text{det}\begin{bmatrix} -1 & -1 \\ 1 & 2 \end{bmatrix} = -((-1 \cdot 2) - (-1 \cdot 1)) = -(-2 + 1) = 1 \]

3. **Cofactor \( C_{13} \)**:
\[ C_{13} = \text{det}\begin{bmatrix} -1 & 2 \\ 1 & -1 \end{bmatrix} = (-1 \cdot -1) - (2 \cdot 1) = 1 - 2 = -1 \]

4. **Cofactor \( C_{21} \)**:
\[ C_{21} = -\text{det}\begin{bmatrix} -1 & 1 \\ -1 & 2 \end{bmatrix} = -((-1 \cdot 2) - (1 \cdot -1)) = -(-2 + 1) = 1 \]

5. **Cofactor \( C_{22} \)**:
\[ C_{22} = \text{det}\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} = (2 \cdot 2) - (1 \cdot 1) = 4 - 1 = 3 \]

6. **Cofactor \( C_{23} \)**:
\[ C_{23} = -\text{det}\begin{bmatrix} 2 & -1 \\ 1 & -1 \end{bmatrix} = -((2 \cdot -1) - (-1 \cdot 1)) = -(-2 + 1) = 1 \]

7. **Cofactor \( C_{31} \)**:
\[ C_{31} = \text{det}\begin{bmatrix} -1 & 1 \\ 2 & -1 \end{bmatrix} = (-1 \cdot -1) - (1 \cdot 2) = 1 - 2 = -1 \]

8. **Cofactor \( C_{32} \)**:
\[ C_{32} = -\text{det}\begin{bmatrix} 2 & 1 \\ -1 & -1 \end{bmatrix} = -((2 \cdot -1) - (1 \cdot -1)) = -(-2 + 1) = 1 \]

9. **Cofactor \( C_{33} \)**:
\[ C_{33} = \text{det}\begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} = (2 \cdot 2) - (-1 \cdot -1) = 4 - 1 = 3 \]

#### Cofactor Matrix

The cofactor matrix \( C \) is:

\[ C = \begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \]

#### Adjoint of Matrix A

Taking the transpose of the cofactor matrix:

\[ \text{adj}(A) = C^T = \begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \]

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

Using the formula for the inverse:

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

Substituting the values:

\[ A^{-1} = \frac{1}{4} \begin{bmatrix} 3 & 1 & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix} \]

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

\[ A^{-1} = \begin{bmatrix} \frac{3}{4} & \frac{1}{4} & -\frac{1}{4} \\ \frac{1}{4} & \frac{3}{4} & \frac{1}{4} \\ -\frac{1}{4} & \frac{1}{4} & \frac{3}{4} \end{bmatrix} \]