If A and B are square matrices of order 3 such that `absA=-1,absB=3," then "abs(3AB)` equals

Options

A. `-9`
B. `81`
C. `-27`
D. `81`

Answer

To solve the problem, we need to find the determinant of the matrix \(3AB\) given that \(\text{det}(A) = -1\) and \(\text{det}(B) = 3\). Here are the steps to arrive at the solution:

### Step-by-Step Solution:

1. **Understanding the Determinant of a Scalar Multiple of a Matrix**:
For any square matrix \(X\) of order \(n\) and a scalar \(k\), the determinant of the matrix \(kX\) is given by:
\[ \text{det}(kX) = k^n \cdot \text{det}(X) \]
Since \(A\) and \(B\) are \(3 \times 3\) matrices, we have \(n = 3\).

2. **Applying the Formula to \(3AB\)**:
We need to find \(\text{det}(3AB)\). According to the properties of determinants:
\[ \text{det}(3AB) = 3^3 \cdot \text{det}(AB) \]

3. **Finding \(\text{det}(AB)\)**:
The determinant of the product of two matrices is the product of their determinants:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
Substituting the given values:
\[ \text{det}(AB) = (-1) \cdot 3 = -3 \]

4. **Calculating \(\text{det}(3AB)\)**:
Now substituting \(\text{det}(AB)\) back into the equation for \(\text{det}(3AB)\):
\[ \text{det}(3AB) = 3^3 \cdot (-3) = 27 \cdot (-3) = -81 \]

### Final Answer:
Thus, the value of \(\text{det}(3AB)\) is \(-81\).

---