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$A$ and $B$ are $n\times n$ matrices.

Any hints on how to solve this or where to find the answer are welcome

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    $\begingroup$ Hint: There exists a theorem that relates the rank of matrices with the rank of their product. $\endgroup$
    – Git Gud
    Jan 27, 2014 at 1:04

2 Answers 2

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Hint: The rank of a product is always less or equal to the rank of each factor.

Also what is the maximum rank of a $n \times n$ matrix?.

Now you should be able to figure this out

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If an $n\times n$ matrix $C$ has rank $n$ it is invertible, which means $\mathrm{det}(C)\neq 0$. If the rank is less than $n$ then $\mathrm{det}(C)=0$. Since $\mathrm{det}(AB)=\mathrm{det}(A)\mathrm{det}(B)$, we can only have $\mathrm{det}(AB)\neq 0$ if $\mathrm{det}(A)\neq 0$ and $\mathrm{det}(B)\neq 0$, so therefore $A$ and $B$ must have rank $n$.

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