The inverse of `[[0, 0, 1], [0, 1, 0], [1, 0, 0]]` is

Options

A. `[[1, 0, 0], [0, 1, 0], [0, 0, 1]]`
B. `[[0, 0, -1], [0, -1, 0], [0, 0, -1]]`
C. `[[0, 0, 1], [0, 1, 0], [1, 0, 0]]`
D. `[[-1, 0, 0], [0, 1, 0], [0, 0, 1]]`

Answer

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

### Step 1: Calculate the Determinant of Matrix A
To find the inverse of a matrix, we first need to calculate its determinant. The determinant of a 3x3 matrix can be calculated using the formula:

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

For our matrix \( A \):

\[ A = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \]

We can expand the determinant along the first row:

\[ \text{det}(A) = 0 \cdot (1 \cdot 0 - 0 \cdot 0) - 0 \cdot (0 \cdot 0 - 1 \cdot 1) + 1 \cdot (0 \cdot 0 - 1 \cdot 1) \]

Calculating this gives:

\[ \text{det}(A) = 0 - 0 + 1 \cdot (-1) = -1 \]

### Step 2: Check if the Determinant is Non-Zero
Since \( \text{det}(A) = -1 \) (which is not zero), the inverse of the matrix exists.

### Step 3: Calculate the Cofactor Matrix
Next, we need to find the cofactor matrix \( C \). The cofactor \( C_{ij} \) is calculated by taking the determinant of the submatrix that remains after removing the \( i \)-th row and \( j \)-th column, multiplied by \( (-1)^{i+j} \).

Calculating the cofactors:

- \( C_{11} = \text{det}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = 0 \)
- \( C_{12} = \text{det}\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = 0 \)
- \( C_{13} = \text{det}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = -1 \)

- \( C_{21} = \text{det}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = -1 \)
- \( C_{22} = \text{det}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = 0 \)
- \( C_{23} = \text{det}\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = 0 \)

- \( C_{31} = \text{det}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = 0 \)
- \( C_{32} = \text{det}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = 0 \)
- \( C_{33} = \text{det}\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = -1 \)

Thus, the cofactor matrix \( C \) is:

\[ C = \begin{bmatrix} 0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]

### Step 4: Transpose the Cofactor Matrix to Get the Adjoint
The adjoint of matrix \( A \) is the transpose of the cofactor matrix:

\[ \text{adj}(A) = C^T = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & 0 \\ -1 & 0 & -1 \end{bmatrix} \]

### Step 5: Calculate the Inverse of Matrix A
Finally, we can find the inverse of matrix \( A \) using the formula:

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

Substituting the values we have:

\[ A^{-1} = \frac{1}{-1} \cdot \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & 0 \\ -1 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{bmatrix} \]

### Final Answer
The inverse of the matrix \( A \) is:

\[ A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} \]