If `A^(-1)=(1)/(3)[[1, 4, -2], [-2, -5, 4], [1, -2, 1]] and |A|=3`, then adjA=

Options

A. `(1)/(9)[[1, 4, -2], [-2, -5, 4], [1, -2, 1]]`
B. `[[1, -2, 1], [4, -5, -2], [-2, 4, 1]]`
C. `[[1, 4, -2], [-2, -5, 4], [1, -2, 1]]`
D. `[[-1, -4, -2], [2, 5, 4], [1, -4, 1]]`

Answer

To find the adjoint of matrix \( A \) given that \( A^{-1} = \frac{1}{3} \begin{bmatrix} 1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1 \end{bmatrix} \) and \( |A| = 3 \), we can use the relationship between the inverse of a matrix and its adjoint.

### Step-by-Step Solution:

1. **Recall the formula for the inverse of a matrix**:
\[ A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A) \]
where \( |A| \) is the determinant of \( A \) and \( \text{adj}(A) \) is the adjoint of \( A \).

2. **Rearranging the formula to find the adjoint**:
From the formula, we can express the adjoint as:
\[ \text{adj}(A) = |A| \cdot A^{-1} \]

3. **Substituting the known values**:
We know that \( |A| = 3 \) and \( A^{-1} = \frac{1}{3} \begin{bmatrix} 1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1 \end{bmatrix} \).
Thus, we can substitute these values into the equation:
\[ \text{adj}(A) = 3 \cdot \left( \frac{1}{3} \begin{bmatrix} 1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1 \end{bmatrix} \right) \]

4. **Simplifying the expression**:
The multiplication simplifies as follows:
\[ \text{adj}(A) = \begin{bmatrix} 1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1 \end{bmatrix} \]

5. **Final result**:
Therefore, the adjoint of matrix \( A \) is:
\[ \text{adj}(A) = \begin{bmatrix} 1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1 \end{bmatrix} \]