What is the inverse of `[{:(0,0,1),(0,1,0),(1,0,0):}]` ?

Options

A. `[{:(1,0,0),(0,1,0),(0,0,1):}]`
B. `[{:(0,0,1),(0,1,0),(1,0,0):}]`
C. `[{:(-1,0,1),(0,-1,0),(0,0,-1):}]`
D. `[{:(0,0,1),(0,-1,0),(-1,0,0):}]`

Answer

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

### Step 1: Define the Matrix
Let \( A = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \).

### Step 2: Find the Determinant of A
To find the inverse, we first need to calculate the determinant of matrix \( A \).

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

### Step 3: Check if the Determinant is Zero
Since the determinant is \( 0 \), the matrix \( A \) is singular, which means it does not have an inverse.

### Conclusion
Thus, the inverse of the matrix \( A \) does not exist.

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