Question

Let \( A \) be a square matrix of order 3 and \( |A| \) denote the determinant of \( A \). If \( A = kB \), where \( k \) is a scalar, then \( |A| \) is equal to

• A. \( |B| \)
• B. \( 3|B| \)
• C. \( k|B| \)
• D. \( k^3|B| \)

Hint:

For a square matrix of order \( n \), if the matrix is multiplied by a scalar \( k \), the determinant is multiplied by \( k^n \).

Answer 1

The Correct Option is D

Step 1: Determinant properties.
When a matrix \( A \) is equal to a scalar \( k \) multiplied by matrix \( B \), i.e., \( A = kB \), the determinant of \( A \) is related to the determinant of \( B \) by the property: \[ |A| = |kB| = k^n |B| \] where \( n \) is the order of the square matrix. For a 3x3 matrix, \( n = 3 \).
Step 2: Conclusion.
Thus, \( |A| = k^3 |B| \). Therefore, the correct answer is (4) \( k^3|B| \).