Question

Let A and B be matrices of order 3 × 3. If |AB| = 0, then which of the following can be concluded?
1. A = 0 and B = 0
2. |A| = 0 and |B| = 0
3. Either |A| = 0 or |B| = 0
4. Either A = 0 or B = 0

Answer 1

Correct Answer - Option 3 : Either |A| = 0 or |B| = 0

Concept:

Let A and B be any two square matrices then |AB| = |A||B|.

Calculations:

Given, A and B be matrices of order 3 × 3.

|AB| = 0,

We know that |AB| = |A||B|

⇒ |A||B| = 0

⇒ either |A| = 0 or |B| = 0 or both |A| = |B| = 0.

⇒ either |A| = 0 or |B| = 0 .

Hence, if A and B be matrices of order 3 × 3. If AB = 0, either |A| = 0 or |B| = 0.