Matrices: $AB=0 \implies A=0 \text{ or } \ B=0$

Question

When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:- $$AB=0 \implies A=0 \text{ or } \ B=0$$

I proved it as follows:-

Assume $A \neq O$ and $ B \neq O$: then, $$ |A||B| \neq 0 $$ $$ |AB| \neq 0 $$ $$ AB \neq O $$ $$ \therefore A \neq O\ and\ B \neq O \implies AB \neq O $$ $$ \neg[ AB \neq O] \implies \neg [ A \neq O\ and\ B \neq O ] $$ $$AB=O \implies A=O \text{ or } \ B=O$$

But when considering, A := \begin{pmatrix} 1&1 \\1&1 \end{pmatrix} and B:= \begin{pmatrix} -1& 1\\ 1 &-1 \end{pmatrix}then, AB=O and A$\neq $O and B $\neq$ O

I can't figure out which one and where I went wrong.

Why are you using $O$ and $0$ differently? What do these mean to you? — Thomas Andrews, Jul 19 '14 at 13:59
If $|A|$ is the dterminant, then it is not true that $A\neq O\implies |A|\neq 0$. — Thomas Andrews, Jul 19 '14 at 14:00
Git Gud has already pointed out your mistake. Check the case of your own example, both $A$ and $B$ have determinant $0$. — M. Vinay, Jul 19 '14 at 14:05

DGRasines · Accepted Answer · 2016-02-23T15:08:10.000

3

You are saying that if $A \neq O$ then, $det(A) \neq O$, which is false in general. Consider any diagonal matrix different from $O$ which has at least one zero in the diagonal.

edited Feb 23 '16 at 15:08

answered Jul 19 '14 at 14:08

DGRasines

533

score 0 · Answer 2 · answered Feb 22 '16 at 14:35

0

You aren't wrong! AB = O does not imply that either A or B is zero. In matrices there is no such case. So even when A is not equal to O and B is not equal to zero, you can get AB =O. Voila!

answered Feb 22 '16 at 14:35

Barath

1

1

This old question already has an accepted answer. What's the point of your answer? – Silvia Ghinassi Feb 22 '16 at 14:56

score 0 · Answer 3 · answered Jul 19 '14 at 14:12

0

The problem with your first approach is that there exist non-zero matrices with zero determinant. For instance, $A$ and $B$ in your example.

answered Jul 19 '14 at 14:12

Marco Flores

2,769

Matrices: $AB=0 \implies A=0 \text{ or } \ B=0$

3 Answers3