If A and B are squar matrices of order 3 such that |A|=-1, |B|=3 then |3AB| is equal to

Options

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

Answer

To find the determinant of the matrix \(3AB\) where \(A\) and \(B\) are square matrices of order 3, we can use the properties of determinants.

### Step-by-Step Solution:

1. **Understanding the Determinant of a Scalar Multiple of a Matrix**:
The determinant of a scalar multiple of a matrix can be expressed as:
\[ |kA| = k^n |A| \]
where \(k\) is a scalar, \(A\) is a matrix, and \(n\) is the order of the matrix.

2. **Applying the Formula**:
In our case, we have \(k = 3\) and the order \(n = 3\) (since \(A\) and \(B\) are both 3x3 matrices). Therefore, we can write:
\[ |3AB| = 3^3 |AB| = 27 |AB| \]

3. **Finding the Determinant of the Product of Two Matrices**:
The determinant of the product of two matrices is given by:
\[ |AB| = |A| \cdot |B| \]
Given that \(|A| = -1\) and \(|B| = 3\), we can calculate:
\[ |AB| = |A| \cdot |B| = (-1) \cdot 3 = -3 \]

4. **Substituting Back to Find |3AB|**:
Now we substitute \(|AB|\) back into our earlier expression:
\[ |3AB| = 27 |AB| = 27 \cdot (-3) = -81 \]

5. **Final Result**:
Thus, the determinant \(|3AB|\) is:
\[ |3AB| = -81 \]

### Conclusion:
The value of \(|3AB|\) is \(-81\).