Q. If A and B are two non-zero square matrices of the same order such that the product $AB = O$, then
A) Both A and B must be singular
B) Exactly one of them must be singular
C) Both of them are non-singular
D) None of these
Explanation
(b)
If possible assume that $A$ is non-singular, then $A^{- 1}$ exists.
Thus,
$AB = 0 \Rightarrow A^{- 1}(AB) = \left( A^{- 1}A \right)B = 0$
$\Rightarrow IB = 0\$or$\ B = 0 \times$(a contradiction)
Hence, both $A$ and $B$ must be singular.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.