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 ?
Best Answer
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 $\operatorname{rk} AB = \dim \operatorname{im} AB = \dim \operatorname{im} A = \operatorname{rk} A$.