Question

Let A = [aij]2x3 and B = [bij]3x2, then |5AB| is equal to

• A. \( 5^2 . |A| \cdot |B| \)
• B. \( 5^3 \cdot |A| \cdot |B| \)
• C. 5^{2} |AB|
• D. 5^{3} |AB|

Hint:

Two Crucial Rules to Remember:

• The order of a product of matrices \( (m \times n) \) * \( (n \times p) \) is \( (m \times p) \).
• The determinant of a scalar multiple \( |kM| \) is \( k^n |M| \), where 'n' is the order of the square matrix \( M \), not necessarily the scalar's exponent in the original expression.

Answer 1

The Correct Option is C

Step 1: Conceptual Understanding:

This problem requires knowledge of matrix multiplication properties and determinants. The objective is to identify the dimensions of the product matrix \( AB \) and then apply the determinant property for scalar multiples of a matrix.

Step 2: Key Principles:

1. Determine the dimensions of the product matrix \( AB \).
2. Apply the determinant property: For a square matrix \( M \) of order \( n \) and a scalar \( k \), the determinant of \( kM \) is \( |kM| = k^n |M| \).

Step 3: Detailed Breakdown:

First, ascertain the dimensions of the product matrix \( AB \).

Matrix \( A \) has dimensions \( 2 \times 3 \).
Matrix \( B \) has dimensions \( 3 \times 2 \).
Matrix multiplication \( AB \) is valid if the number of columns in \( A \) equals the number of rows in \( B \). This condition is met (3).
The dimensions of the resulting matrix \( AB \) are (rows of \( A \)) \( \times \) (columns of \( B \)), resulting in a \( 2 \times 2 \) matrix.
Therefore, \( AB \) is a square matrix with order \( n = 2 \).

Next, calculate the determinant of \( 5AB \) using the property \( |kM| = k^n |M| \).

In this case, \( M = AB \), \( k = 5 \), and \( n = 2 \).

\[ |5AB| = 5^2 |AB| = 25 |AB| \]

Note: Options (1) and (2) are incorrect as determinants are defined only for square matrices. Since \( A \) (\( 2 \times 3 \)) and \( B \) (\( 3 \times 2 \)) are not square, their individual determinants \( |A| \) and \( |B| \) are undefined.

Step 4: Conclusion:

The value of \( |5AB| \) is \( 5^2 |AB| \).