Rank of matrices

Question

If A is an $(m * k)$ - matrix and B is a $(k * n)$ - matrix, how would I show that $rnk(AB) ≤ rnk(B)$? I'm confused as to what conditions on matrix A would ensure that this equality holds. What would they be?

Answer 1

every row of $AB$ is a linear combination of the rows of $B.$ that means the row space of $AB$ is contained within the row space of $B.$ therefore the rank of $AB$ which is the dimension of the row space is less or equal to the rank of $B.$

Answer 2

$$ Bx = 0 \Rightarrow (AB) x = A(Bx) = A0 = 0 $$ This means $$ \mbox{dim } \mbox{ker } B \le \mbox{dim } \mbox{ker } AB $$ The rank nullity theorem gives: \begin{align} \mbox{rk }AB &= n -\mbox{dim } \mbox{ker } AB \\ \mbox{rk } B &= n -\mbox{dim } \mbox{ker } B \end{align}

This gives $$ \mbox{rk } AB = \mbox{rk } B + \mbox{dim } \mbox{ker } B - \mbox{dim } \mbox{ker } AB \le \mbox{rk } B $$