A and B are two matrices of same order `3 xx 3`, where `A=[{:(1,2,3),(2,3,4),(5,6,8):}],B=[{:(3,2,5),(2,3,8),(7,2,9):}]`
The value of |adj (AB)| is

Options

A. 24
B. `24^2`
C. `24^3`
D. `65`

Answer

To find the value of |adj(AB)| for the given matrices A and B, we can follow these steps:

### Step 1: Define the matrices A and B
Given:
\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 2 & 5 \\ 2 & 3 & 8 \\ 7 & 2 & 9 \end{pmatrix} \]

### Step 2: Calculate the determinant of AB
To find |adj(AB)|, we first need to calculate the determinant of the product of matrices A and B. The formula for the determinant of the product of two matrices is:
\[ |AB| = |A| \cdot |B| \]

#### Step 2.1: Calculate |A|
Using the determinant formula for a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where:
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
For matrix A:
\[ |A| = 1(3 \cdot 8 - 4 \cdot 6) - 2(2 \cdot 8 - 4 \cdot 5) + 3(2 \cdot 6 - 3 \cdot 5) \]
Calculating each term:
- First term: \(1(24 - 24) = 0\)
- Second term: \(-2(16 - 20) = -2(-4) = 8\)
- Third term: \(3(12 - 15) = 3(-3) = -9\)

Thus:
\[ |A| = 0 + 8 - 9 = -1 \]

#### Step 2.2: Calculate |B|
Using the same determinant formula for matrix B:
\[ |B| = 3(3 \cdot 9 - 8 \cdot 2) - 2(2 \cdot 9 - 8 \cdot 7) + 5(2 \cdot 2 - 3 \cdot 7) \]
Calculating each term:
- First term: \(3(27 - 16) = 3 \cdot 11 = 33\)
- Second term: \(-2(18 - 56) = -2(-38) = 76\)
- Third term: \(5(4 - 21) = 5(-17) = -85\)

Thus:
\[ |B| = 33 + 76 - 85 = 24 \]

### Step 3: Calculate |AB|
Now we can find |AB|:
\[ |AB| = |A| \cdot |B| = (-1) \cdot 24 = -24 \]

### Step 4: Calculate |adj(AB)|
The property of the adjoint states that:
\[ |adj(AB)| = |AB|^{n-1} \]
where \(n\) is the order of the matrix. For a 3x3 matrix, \(n = 3\):
\[ |adj(AB)| = |AB|^{3-1} = |AB|^2 = (-24)^2 = 576 \]

### Final Answer
The value of |adj(AB)| is:
\[ \boxed{576} \]