[Math] About the relation of rank(AB), rank(A), rank(B) and the zero matrix

linear algebra

Let $A$ be a $2 \times 4$ matrix and $B$ be a $4 \times 4$ matrix, prove that if $rank(A) = 2$ and $rank(B)=3$ then $AB \neq 0$.

I got stuck at $rank(AB) \leq 2 $

How do I continue from here?

Best Answer

As pointed out in the comments your matrix $A$ is the wrong way round... As, for our purposes, is the inequality! (It's right, just not helpful since what we want is $\mathrm{rank}(AB)\ge1$.)

Here's an equivalent way of saying $AB=0$.

The image of $B$ is entirely contained within the kernel of $A$.

Can you see why this can't happen?

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