Question

A and B and n x n matrices such that AB = 0. Prove that if A is invertible then B is not invertible.

Answer 1

Answer and Step-by-step explanation:

Since we have given that

AB = 0

where A and B are n x n matrices.

Consider determinant on both sides,

`\mid AB\mid=\mid 0\mid\\\\\mid A\mid \mid B\mid =0\\\\either\ \mid A\mid =0\ or\ \mid B\mid =0`

since A is invertible, then |A| ≠ 0

so, it means |B| = 0.

Hence, B is not invertible.

Hence proved.