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):}]` Value of |adj (adj adj( adj A)))| is

Options

A. `2^4`
B. `2^9`
C. `1`
D. `2^19`

Answer

To find the value of \(|\text{adj}(\text{adj}(\text{adj}(\text{adj} A)))|\), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A

Given the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{pmatrix} \), we need to find the determinant \(|A|\).

Using the determinant formula for a \(3 \times 3\) 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 our matrix:
- \( a = 1, b = 2, c = 3 \)
- \( d = 2, e = 3, f = 4 \)
- \( g = 5, h = 6, i = 8 \)

Calculating the determinant:
\[ |A| = 1(3 \cdot 8 - 4 \cdot 6) - 2(2 \cdot 8 - 4 \cdot 5) + 3(2 \cdot 6 - 3 \cdot 5) \]
\[ = 1(24 - 24) - 2(16 - 20) + 3(12 - 15) \]
\[ = 1(0) - 2(-4) + 3(-3) \]
\[ = 0 + 8 - 9 \]
\[ = -1 \]

### Step 2: Use the Property of Determinants of Adjoint Matrices

The property of determinants states that:
\[ |\text{adj}(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix. For a \(3 \times 3\) matrix, \( n = 3 \).

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

### Step 3: Calculate the Determinant of the Adjoint of the Adjoint

Now, we need to find \(|\text{adj}(\text{adj}(A))|\):
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{3-1} = |A|^{(3-1)(3-1)} = |A|^{2} = 1^2 = 1 \]

### Step 4: Calculate the Determinant of the Adjoint of the Adjoint of the Adjoint

Next, we find \(|\text{adj}(\text{adj}(\text{adj}(A)))|\):
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = |\text{adj}(\text{adj}(A))|^{3-1} = |A|^{(3-1)(3-1)} = |A|^{2} = 1^2 = 1 \]

### Step 5: Calculate the Determinant of the Adjoint of the Adjoint of the Adjoint of the Adjoint

Finally, we find \(|\text{adj}(\text{adj}(\text{adj}(\text{adj}(A))))|\):
\[ |\text{adj}(\text{adj}(\text{adj}(\text{adj}(A))))| = |\text{adj}(\text{adj}(\text{adj}(A)))|^{3-1} = |A|^{(3-1)(3-1)} = |A|^{2} = 1^2 = 1 \]

### Conclusion

Thus, the value of \(|\text{adj}(\text{adj}(\text{adj}(\text{adj}(A))))|\) is \( \boxed{1} \).