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

Answer

To find the value of \(|3(\text{adj } A)BC^{-1}|\), we can follow these steps:

### Step 1: Use the property of determinants
We know that for any scalar \(k\) and a square matrix \(M\) of order \(n\):
\[ |kM| = k^n |M| \]
Here, \(k = 3\) and \(n = 3\) (since \(A\), \(B\), and \(C\) are \(3 \times 3\) matrices). Therefore,
\[ |3(\text{adj } A)BC^{-1}| = 3^3 |\text{adj } A| |B| |C^{-1}| \]

### Step 2: Calculate \(3^3\)
Calculating \(3^3\):
\[ 3^3 = 27 \]

### Step 3: Find \(|\text{adj } A|\)
Using the property of the adjoint, we have:
\[ |\text{adj } A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. Since \(|A| = 2\) and \(n = 3\):
\[ |\text{adj } A| = |A|^{3-1} = |A|^2 = 2^2 = 4 \]

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

### Step 5: Combine all parts
Now substituting back into our expression:
\[ |3(\text{adj } A)BC^{-1}| = 27 |\text{adj } A| |B| |C^{-1}| \]
Substituting the values we have:
\[ = 27 \cdot 4 \cdot 3 \cdot \frac{1}{4} \]

### Step 6: Simplify the expression
The \(4\) in the numerator and denominator cancels out:
\[ = 27 \cdot 3 = 81 \]

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

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