Let and be non-zero real matrices such that

MATHEMATICS · Subjective · CBSE · Mathematics · Matrices · Matrices

Problem

Let

A

and

B

be non-zero

2\times 2

real matrices such that

AB=O_{2\times 2}

. (a) Prove that at least one of the matrices

A

B

must be singular. (b) Further prove that, if both

A

and

B

are non-zero and

AB=O_{2\times 2}

, then each of

A

and

B

is necessarily singular. (c) Give an explicit example of two non-zero singular

2\times 2

matrices whose product is the zero matrix and verify your claim by direct multiplication.