Rank of an n x n matrix A and how it relates to rank $A^2$

Question

In an old exam, I saw that if A is a 5 x 5 matrix with rank(A) = 3, then rank($A^2$) $\geq 1$. Why is this true?

Answer 1

As said in comments, you can argue by contradiction. Assume that $A^2x=0,\forall x\in V,$ with $\dim V=5.$ So, it is

$$A(Ax)=0, \forall x\in V.$$ That is,

$$Image(A)\subset \ker(A).$$ Thus $$\dim(Image(A))\le \dim (\ker(A)).$$ Moreover $$\dim(Image(A))+\dim(\ker(A))=5.$$ So, it must be $\dim(\ker A)\ge 3$ which contradicts the fact that $rank(A)=3.$ So, it must be $rank(A^2)\ge 1.$