TITLE: How to give counterexample for given claim
QUESTION [5 upvotes]: Suppose $A_1,\dots,A_m$ be distinct $n\times n $ real matrices such that $A_iA_j=0$ for all $i\neq j$. Show that $m\leq n$.

I think this true because i tried for $3\times 3$ and $2\times 2$ case I got only $3$ and $2$ matrices with that property.
But given that this not true .
Can any one help me to find counterexample.
And what is best approach to tackle such problem.

REPLY [7 votes]: This is false. Let $A$ be a $2\times 2$ matrix such that $A^{2}=0$. The the collection $\{cA:c\in \mathbb R\}$ has this property.