If A= `[ (1,2,-1),(-1,1,2),(2,-1,1)]` then det {adj (A)} equals

Options

A. `(14)^(2)`
B. `(13)^(2)`
C. `(14)^(3)`
D. `(13)^(3)`

Answer

To find the determinant of the adjoint of matrix \( A \), we can use the property that states:

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

where \( n \) is the order of the matrix \( A \).

### Step 1: Identify the matrix \( A \)
Given:
\[ A = \begin{pmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{pmatrix} \]

### Step 2: Calculate the determinant of matrix \( A \)
To calculate \( \text{det}(A) \), we can use the formula for the determinant of a 3x3 matrix:

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

Where:
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

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

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

Calculating each term:
1. \( 1(1 + 2) = 1 \cdot 3 = 3 \)
2. \( -2(-1 - 4) = -2 \cdot (-5) = 10 \)
3. \( -1(1 - 2) = -1 \cdot (-1) = 1 \)

Putting it all together:
\[ \text{det}(A) = 3 + 10 + 1 = 14 \]

### Step 3: Determine the order of the matrix \( A \)
The matrix \( A \) is a \( 3 \times 3 \) matrix, so \( n = 3 \).

### Step 4: Calculate \( \text{det}(\text{adj}(A)) \)
Using the property mentioned earlier:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} = (14)^{3-1} = (14)^{2} = 196 \]

### Final Answer
\[ \text{det}(\text{adj}(A)) = 196 \]

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