Question

If \( A \) is a square matrix of order 3 such that its determinant is \( |A| = 3 \), calculate the value of the scalar matrix determinant represented by \( |2A| \).

• A. \( 6 \)
• B. \( 24 \)
• C. \( 12 \)
• D. \( 18 \)

Hint:

Remember to always check the order of the matrix (\( n \)) before scaling a determinant. Forgetting to raise the scalar factor to the power of \( n \) is a very common exam mistake.

Answer 1

The Correct Option is B

Concept: For any square matrix \( A \) of order \( n \times n \), factoring out a scalar multiplier \( k \) from inside a determinant requires raising that scalar to the power of the matrix order: \[ |kA| = k^n |A| \] This occurs because the scalar multiplier scales every individual row of the matrix uniformly.

Step 1: Identify the scale factor, matrix order, and base determinant.
The problem provides the following parameters:

• Scalar factor (\( k \)) = 2
• Matrix order (\( n \)) = 3
• Base determinant (\( |A| \)) = 3

Step 2: Apply parameters to the determinant scaling property.
Substitute these values into the scaling identity: \[ |2A| = 2^3 \times |A| \]

Step 3: Simplify the numerical expression.
Evaluate the cubic exponent and multiply by the base determinant: \[ |2A| = 8 \times 3 = 24 \]