If `A=[(1,2,-1),(-1,1,2),(2,-1,1)]` then det `[adj(adjA)]=`

Options

A. `12^(4)`
B. `13^(4)`
C. `14^(4)`
D. None of these

Answer

To solve the problem, we need to find the determinant of the adjoint of the adjoint of matrix \( A \). The matrix \( A \) is given as:

\[ A = \begin{pmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{pmatrix} \]

### Step 1: Calculate the Determinant of \( A \)

We can calculate the determinant of \( A \) using the formula for the determinant of a \( 3 \times 3 \) matrix:

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

Where:
- \( a, b, c \) are the elements of the first row,
- \( d, e, f \) are the elements of the second row,
- \( g, h, i \) are the elements of the third row.

For our matrix \( A \):

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

Calculating each term:

- First term: \( 1 \cdot (1 + 2) = 1 \cdot 3 = 3 \)
- Second term: \( -2 \cdot (-1 - 4) = -2 \cdot (-5) = 10 \)
- Third term: \( -1 \cdot (1 - 2) = -1 \cdot (-1) = 1 \)

Putting it all together:

\[ \text{det}(A) = 3 + 10 + 1 = 14 \]

### Step 2: Use the Formula for the Determinant of the Adjoint

The formula for the determinant of the adjoint of a matrix \( A \) is given by:

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

Where \( n \) is the order of the matrix. Since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \).

Thus,

\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{3-1} = (\text{det}(A))^2 \]

### Step 3: Calculate the Determinant of the Adjoint of \( A \)

Now substituting the value of \( \text{det}(A) \):

\[ \text{det}(\text{adj}(A)) = (14)^{2} = 196 \]

### Step 4: Calculate the Determinant of the Adjoint of the Adjoint of \( A \)

Again applying the formula for the adjoint:

\[ \text{det}(\text{adj}(\text{adj}(A))) = (\text{det}(\text{adj}(A)))^{n-1} = (\text{det}(\text{adj}(A)))^{2} \]

Substituting the value we found:

\[ \text{det}(\text{adj}(\text{adj}(A))) = (196)^{2} = 38416 \]

### Final Answer

Thus, the final answer is:

\[ \text{det}(\text{adj}(\text{adj}(A))) = 38416 \]