Question

If \( A \) is a square matrix of order 3 such that its determinant value is \( |A| = -2 \), find the value of the scalar-scaled determinant \( |4A| \).

• A. \( -128 \)
• B. \( -8 \)
• C. \( -24 \)
• D. \( 128 \)

Hint:

Never multiply a determinant directly by a scalar without checking the matrix order first. The scalar multiplier must always be raised to the power of the order (\( k^n \)) before completing the calculation.

Answer 1

The Correct Option is A

Concept: For any square matrix \( A \) of order \( n \) and a scalar multiplier constant \( k \), pulling the scalar factor outside the determinant brackets requires raising it to the power of the matrix order: \[ |k \cdot A| = k^n \cdot |A| \]

Step 1: Identify the order and the scalar factor from the problem. From the problem statement, we isolate the key metrics:
• Matrix order, \( n = 3 \)
• Scalar multiplier factor, \( k = 4 \)
• Original determinant value, \( |A| = -2 \)

Step 2: Apply the scalar scaling property formula. Substitute our values directly into the property formula: \[ |4A| = 4^3 \cdot |A| \] Calculate the value of the cubed scalar term: \[ 4^3 = 4 \cdot 4 \cdot 4 = 64 \]

Step 3: Compute the final determinant value. Multiply the expanded scalar by the original determinant value: \[ |4A| = 64 \cdot (-2) = -128 \]