Q:

Column Space of a matrix contained in Null Space of other matrix

So I have $CS(A)$ is the subset of $NS(B)$. How do I show that $(AB)^2$ is zero matrix. $A$ and $ B$ are $n$ by $n$ matrices.

A:

Consider any vector $x.$ Then $ABx\in \rm{CS}(A).$ Since $\rm{CS}(A)\subset \rm{NS}(B)$ we have that $BABx=0.$ So
$$(AB)^2x=ABABx=A0=0.$$ Since $x$ is arbitrary we conclude that $(AB)^2=0.$

