Question

If A and B are matrices of order 3 and |A| = 5, |B| = 3 then |3AB| is

• A. 425
• B. 405
• C. 565
• D. 585

Answer 1

The Correct Option is B

Given:
\( A \) and \( B \) are matrices of order 3.
\( |A| = 5 \), \( |B| = 3 \).
We are asked to find \( |3ABI| \).

According to the properties of determinants, we know: \[ |kA| = k^n |A| \] where \( k \) is a scalar and \( n \) is the order of the matrix.

Since \( A \) and \( B \) are 3x3 matrices, \( n = 3 \). Therefore:

\[ |3A| = 3^3 |A| = 27 \times 5 = 135 \] We also know that \( |ABI| = |A| \times |B| \times |I| \), where \( |I| = 1 \). So: \[ |ABI| = |A| \times |B| = 5 \times 3 = 15 \] Now, we need to calculate \( |3ABI| \). We know: \[ |3ABI| = |3A| \times |B| \times |I| = 135 \times 3 = 405 \]

Thus, the correct answer is (B): 405.

 

Answer 2

Given that A and B are matrices of order 3, \(|A| = 5\), and \(|B| = 3\).

We need to find \(|3AB|\).

We know that for matrices of order n, if a scalar k is multiplied by a matrix A, then \(|kA| = k^n |A|\).

Also, we know that \(|AB| = |A| |B|\).

Therefore, \(|3AB| = 3^3 |AB| = 27 |A| |B|\).

Substituting the given values, we get \(|3AB| = 27 \times 5 \times 3 = 27 \times 15 = 405\).

Therefore, \(|3AB| = 405\).