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 can use the properties of determinants. Let's break this down step by step.

### Step 1: Use the property of determinants for scalar multiplication
We know that for any scalar \(k\) and an \(n \times n\) matrix \(M\), the determinant can be expressed as:
\[ |kM| = k^n |M| \]
Here, \(k = 3\) and the order of the matrix \(n = 3\). Therefore,
\[ |3(\text{adj} A)BC^{-1}| = 3^3 |\text{adj} A \cdot B \cdot C^{-1}| \]
Calculating \(3^3\):
\[ 3^3 = 27 \]
So we have:
\[ |3(\text{adj} A)BC^{-1}| = 27 |\text{adj} A \cdot B \cdot C^{-1}| \]

### Step 2: Use the property of determinants for the adjoint
The determinant of the adjoint of a matrix \(A\) is given by:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is a \(3 \times 3\) matrix, we have:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 \]
Given that \(|A| = 2\):
\[ |\text{adj} A| = 2^2 = 4 \]

### Step 3: Use the property of determinants for the inverse
For the inverse of a matrix \(C\), the determinant is given by:
\[ |C^{-1}| = \frac{1}{|C|} \]
Given that \(|C| = 4\):
\[ |C^{-1}| = \frac{1}{4} \]

### Step 4: 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:
\[ |\text{adj} A \cdot B \cdot C^{-1}| = 4 \cdot 3 \cdot \frac{1}{4} \]
The \(4\) in the numerator and denominator cancels out:
\[ |\text{adj} A \cdot B \cdot C^{-1}| = 3 \]

### Step 5: Final calculation
Now substituting back into our earlier expression:
\[ |3(\text{adj} A)BC^{-1}| = 27 \cdot 3 = 81 \]

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