If A, B and C are square matrices of order 3 and `|A|=2, |B|=3 and |C|=4`, then the value of `|3(adjA)BC^(-1)|` is equal to (where, adj A represents the adjoint matrix of A)

Options

A. 27
B. `(27)/(4)`
C. `(81)/(2)`
D. 81

Answer

To find the value of \(|3(\text{adj} A)BC^{-1}|\), we will use properties of determinants and the given values of the determinants of matrices \(A\), \(B\), and \(C\).

### Step-by-Step Solution:

1. **Identify the Order of the Matrices**:
Since \(A\), \(B\), and \(C\) are square matrices of order 3, we have \(n = 3\).

2. **Use the Determinant Property**:
The property of determinants states that for any scalar \(k\) and a matrix \(M\) of order \(n\):
\[ |kM| = k^n |M| \]
Therefore, we can write:
\[ |3(\text{adj} A)BC^{-1}| = 3^3 |\text{adj} A \cdot B \cdot C^{-1}| \]

3. **Calculate \(|\text{adj} A|\)**:
The determinant of the adjoint of a matrix \(A\) is given by:
\[ |\text{adj} A| = |A|^{n-1} \]
For our case:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 \]
Given \(|A| = 2\):
\[ |\text{adj} A| = 2^2 = 4 \]

4. **Calculate \(|C^{-1}|\)**:
The determinant of the inverse of a matrix \(C\) is given by:
\[ |C^{-1}| = \frac{1}{|C|} \]
Given \(|C| = 4\):
\[ |C^{-1}| = \frac{1}{4} \]

5. **Combine the Determinants**:
Now we can combine the determinants:
\[ |\text{adj} A \cdot B \cdot C^{-1}| = |\text{adj} A| \cdot |B| \cdot |C^{-1}| \]
Substituting the values we have:
\[ |\text{adj} A \cdot B \cdot C^{-1}| = 4 \cdot |B| \cdot \frac{1}{|C|} \]
Given \(|B| = 3\) and \(|C| = 4\):
\[ |\text{adj} A \cdot B \cdot C^{-1}| = 4 \cdot 3 \cdot \frac{1}{4} = 3 \]

6. **Final Calculation**:
Now substituting back into our earlier expression:
\[ |3(\text{adj} A)BC^{-1}| = 3^3 \cdot 3 = 27 \cdot 3 = 81 \]

### Final Answer:
Thus, the value of \(|3(\text{adj} A)BC^{-1}|\) is \(81\).