Question

Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 = BA, A ≠ 0 and B ≠ 0

Answer 1

Let A = `[(x, - x),(-x, x)] ≠ 0`

B = `[(1, 1),(1, 1)] ≠ 0` 

AB = `[(x, - x),(-x, x)] [(1, 1),(1, 1)]`

= `[(x - x, x - x),(-x +  x, - x + x)]`

AB = `[(0, 0),(0, 0)]`

BA = `[(1, 1),(1, 1)] [(x, -x),(-x, x)]`

BA = `[(x - x, -x + x),(x -  x, - x + x)]`

BA = `[(0, 0),(0, 0)]`

Martices A and B satisfies the condition

AB = 0 = BA

A ≠ 0

B ≠ 0.