If A = `[(1,1,1),(1,2,-3),(2,-1,3)]` then | adj A| =

Options

A. 121
B. 132
C. 178
D. 184

Answer

To find the value of |adj A| for the given matrix \( A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by the formula:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix \( A \):
- \( a = 1, b = 1, c = 1 \)
- \( d = 1, e = 2, f = -3 \)
- \( g = 2, h = -1, i = 3 \)

Now substituting these values into the determinant formula:
\[ |A| = 1 \cdot (2 \cdot 3 - (-3) \cdot (-1)) - 1 \cdot (1 \cdot 3 - (-3) \cdot 2) + 1 \cdot (1 \cdot (-1) - 2 \cdot 2) \]

### Step 2: Simplify Each Term
Calculating each term:
1. First term:
\[ 1 \cdot (6 - 3) = 1 \cdot 3 = 3 \]
2. Second term:
\[ -1 \cdot (3 - (-6)) = -1 \cdot (3 + 6) = -1 \cdot 9 = -9 \]
3. Third term:
\[ 1 \cdot (-1 - 4) = 1 \cdot (-5) = -5 \]

### Step 3: Combine the Results
Now, we combine the results of the three terms:
\[ |A| = 3 - 9 - 5 = -11 \]

### Step 4: Calculate the Determinant of the Adjoint of A
The determinant of the adjoint of a matrix \( A \) is given by the formula:
\[ |\text{adj } A| = |A|^{n-1} \]
where \( n \) is the order of the matrix. Since \( A \) is a 3x3 matrix, \( n = 3 \).

Thus,
\[ |\text{adj } A| = |A|^{3-1} = |A|^2 = (-11)^2 = 121 \]

### Final Answer
\[ |\text{adj } A| = 121 \]

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