If A and B are square matrices each of order n such that |A|=5,|B|=3 and |3AB|=405, then the value of n is :

Options

A. 2
B. 3
C. 4
D. Data insufficient

Answer

To solve the problem, we need to find the value of \( n \) given the determinants of matrices \( A \) and \( B \) and the determinant of \( 3AB \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that \( |A| = 5 \)
- We know that \( |B| = 3 \)
- We also know that \( |3AB| = 405 \)

2. **Using the Property of Determinants:**
- The determinant of a scalar multiple of a matrix is given by the formula:
\[ |kA| = k^n |A| \]
where \( n \) is the order of the square matrix \( A \).

3. **Applying the Property to \( 3AB \):**
- We can express \( |3AB| \) as:
\[ |3AB| = 3^n |AB| \]

4. **Using the Property of the Product of Determinants:**
- The determinant of the product of two matrices is the product of their determinants:
\[ |AB| = |A| \cdot |B| \]
- Therefore, we have:
\[ |AB| = |A| \cdot |B| = 5 \cdot 3 = 15 \]

5. **Substituting into the Determinant of \( 3AB \):**
- Now substituting \( |AB| \) back into our equation for \( |3AB| \):
\[ |3AB| = 3^n \cdot |AB| = 3^n \cdot 15 \]

6. **Setting Up the Equation:**
- We know that \( |3AB| = 405 \), so we can set up the equation:
\[ 3^n \cdot 15 = 405 \]

7. **Solving for \( 3^n \):**
- Dividing both sides by 15:
\[ 3^n = \frac{405}{15} \]
- Calculating the right side:
\[ 3^n = 27 \]

8. **Expressing 27 as a Power of 3:**
- We know that \( 27 = 3^3 \), so we can write:
\[ 3^n = 3^3 \]

9. **Comparing Exponents:**
- Since the bases are the same, we can equate the exponents:
\[ n = 3 \]

### Final Answer:
The value of \( n \) is \( 3 \).