Question

If \(A\) and \(B\) are two square matrices of order \(3\) such that \(|A|=-2,\ |B|=5\), then \(|4AB|=\)

• A. \(-40\)
• B. \(-256\)
• C. \(-640\)
• D. \(-90\)

Hint:

For a matrix of order \(n\), remember \(|kA|=k^n|A|\). Here the order is \(3\), so \(|4AB|=4^3|AB|\).

Answer 1

The Correct Option is C

Concept:
For determinant questions involving matrices, we use the following important properties: \[ |AB|=|A||B| \] Also, if \(A\) is a square matrix of order \(n\), then: \[ |kA|=k^n|A| \] Here, \(A\) and \(B\) are square matrices of order \(3\). Therefore, \(AB\) is also a square matrix of order \(3\).

Step 1: Find \(|AB|\).
Given: \[ |A|=-2 \] and \[ |B|=5 \] Using: \[ |AB|=|A||B| \] we get: \[ |AB|=(-2)(5) \] \[ |AB|=-10 \]

Step 2: Use the determinant property for scalar multiplication.
We need to find: \[ |4AB| \] Since \(AB\) is a matrix of order \(3\), we use: \[ |kM|=k^3|M| \] Here: \[ k=4 \] and \[ M=AB \] So: \[ |4AB|=4^3|AB| \]

Step 3: Substitute \(|AB|=-10\).
\[ |4AB|=4^3(-10) \] \[ =64(-10) \] \[ =-640 \] Hence, the correct answer is: \[ \boxed{(C)\ -640} \]