NDA 2 2024 Maths, question 5

Topic: Matrices and determinants

Q5

Let $A$ and $B$ be two square matrices of same order. If $AB$ is a null matrix, then which one of the following is correct?

(a) Both $A$ and $B$ are null matrices
(b) Either $A$ or $B$ is a null matrix
(c) $B$ is a null matrix if $A$ is a non-singular matrixCorrect answer
(d) Both $A$ and $B$ are singular matrices

Solution.

Answer: (c) $B$ is a null matrix if $A$ is a non-singular matrix

If A is nonsingular, multiply AB=0 by A⁻¹ on the left to obtain B=0.
Nonzero singular matrices can multiply to zero: diag(1,0)diag(0,1)=0.
Also A=I and B=0 disproves the claim that both must be singular.