Find the $rank(AB)$ when rank of $A$ and $B$ are given

Question

Let , $A_{7\times 5}$ be a matrix of rank $3$ and $B_{5\times 7}$ be a matrix of rank $5$. Then find the rank of the matrix $AB$.

As we know , $rank(AB)\le\min\{rank(A),rank(B)\}$ , so $rank(AB)\le 3$. But how I can find out the exact value of $rank(AB)$ ?

Can anyone help me please ?

Answer 1

Hint The rank also satisfies the so called Sylvester’s rank inequality:

If A is an $m \times n$ matrix and B is $n \times k$, then $$rank(AB) \geq rank(A)+ rank(B) -n$$

Answer 2

Consider $A, B$ as linear maps $A:\mathbb{R}^5 \to \mathbb{R}^7$ and $B:\mathbb{R}^7 \to \mathbb{R}^5$. Then, by definition, $\operatorname{rk}(A) = \dim \operatorname{im}(A)$ and $\operatorname{rk}(B) = \dim \operatorname{im}(B)$. In particular, $B$ is surjective. Hence $\operatorname{im} AB = A(\operatorname{im B}) = \operatorname{im}(A)$, and $\operatorname{rk} AB = \dim \operatorname{im} AB = \dim \operatorname{im} A = \operatorname{rk} A$.