The inverse of the matrix `[[1, 2, -2], [0, -2, 1], [-1, 3, 0]]` is

Options

A. `[[3, 6, 2], [1, 2,1], [2, 5, 2]]`
B. `[[3, 6, 2], [10, 2, 1], [2, 5, 2]]`
C. `-[[3, 6, 2], [2, 5, 2], [1, 2, 1]]`
D. `[[2, 5, 2], [3, 6, 2], [1, 2, 1]]`

Answer

To find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 & -2 \\ 0 & -2 & 1 \\ -1 & 3 & 0 \end{bmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:
- \( a = 1, b = 2, c = -2 \)
- \( d = 0, e = -2, f = 1 \)
- \( g = -1, h = 3, i = 0 \)

Calculating the determinant:
\[ \text{det}(A) = 1((-2)(0) - (1)(3)) - 2((0)(0) - (1)(-1)) + (-2)((0)(3) - (-2)(-1)) \]
\[ = 1(0 - 3) - 2(0 + 1) - 2(0 - 2) \]
\[ = -3 - 2 + 4 \]
\[ = -1 \]

### Step 2: Check if the Determinant is Non-Zero
Since \( \text{det}(A) = -1 \neq 0 \), the inverse of the matrix exists.

### Step 3: Calculate the Cofactor Matrix
The cofactor matrix \( C \) is obtained by calculating the cofactors \( C_{ij} \) for each element \( a_{ij} \) of the matrix \( A \).

1. **Cofactor \( C_{11} \)**:
\[ C_{11} = \text{det}\begin{bmatrix} -2 & 1 \\ 3 & 0 \end{bmatrix} = (-2)(0) - (1)(3) = -3 \]

2. **Cofactor \( C_{12} \)**:
\[ C_{12} = -\text{det}\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} = -((0)(0) - (1)(-1)) = -1 \]

3. **Cofactor \( C_{13} \)**:
\[ C_{13} = \text{det}\begin{bmatrix} 0 & -2 \\ -1 & 3 \end{bmatrix} = (0)(3) - (-2)(-1) = -2 \]

4. **Cofactor \( C_{21} \)**:
\[ C_{21} = -\text{det}\begin{bmatrix} 2 & -2 \\ 3 & 0 \end{bmatrix} = -((2)(0) - (-2)(3)) = -6 \]

5. **Cofactor \( C_{22} \)**:
\[ C_{22} = \text{det}\begin{bmatrix} 1 & -2 \\ -1 & 0 \end{bmatrix} = (1)(0) - (-2)(-1) = -2 \]

6. **Cofactor \( C_{23} \)**:
\[ C_{23} = -\text{det}\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix} = -((1)(3) - (2)(-1)) = -5 \]

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

8. **Cofactor \( C_{32} \)**:
\[ C_{32} = -\text{det}\begin{bmatrix} 1 & -2 \\ 0 & 1 \end{bmatrix} = -((1)(1) - (-2)(0)) = -1 \]

9. **Cofactor \( C_{33} \)**:
\[ C_{33} = \text{det}\begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix} = (1)(-2) - (2)(0) = -2 \]

Now, we can write the cofactor matrix \( C \):
\[ C = \begin{bmatrix} -3 & -1 & -2 \\ -6 & -2 & -5 \\ -2 & -1 & -2 \end{bmatrix} \]

### Step 4: Calculate the Adjoint Matrix
The adjoint of matrix \( A \) is the transpose of the cofactor matrix \( C \):
\[ \text{adj}(A) = C^T = \begin{bmatrix} -3 & -6 & -2 \\ -1 & -2 & -1 \\ -2 & -5 & -2 \end{bmatrix} \]

### Step 5: 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:
\[ A^{-1} = \frac{1}{-1} \cdot \begin{bmatrix} -3 & -6 & -2 \\ -1 & -2 & -1 \\ -2 & -5 & -2 \end{bmatrix} = -\begin{bmatrix} -3 & -6 & -2 \\ -1 & -2 & -1 \\ -2 & -5 & -2 \end{bmatrix} \]
\[ = \begin{bmatrix} 3 & 6 & 2 \\ 1 & 2 & 1 \\ 2 & 5 & 2 \end{bmatrix} \]

### Final Answer
The inverse of the matrix \( A \) is:
\[ A^{-1} = \begin{bmatrix} 3 & 6 & 2 \\ 1 & 2 & 1 \\ 2 & 5 & 2 \end{bmatrix} \]