rank of $A^3B^2A$ when $\mathrm{rank}(A)=\mathrm{rank}(B)=n$

Question

Let $A$ and $B$ be two $n\times n$ matrices with rank $n$. Then what is the rank of $C=A^3B^2A$?

I think the answer is $\mathrm{rank}(C)=n$.

I will use the fact that $\mathrm{rank}(AB)=\mathrm{rank}(B)$ when $\mathrm{rank}(A)=n$.

So we get $\mathrm{rank}(C)=\mathrm{rank}(A^3B^2)=\mathrm{rank}(A^3B)=\mathrm{rank}(A^3)=\mathrm{rank}(A^2)=\mathrm{rank}(A)=n$.

I think this is correct . If it is not then any alternative solution humbly welcomed . Thank you .

Answer 1

Yes, it is correct.

Another way to see it is to use

An $n\times n$ matrix has rank $n$ if and only if it is invertible.

Then, using the inverses of $A$ and $B$, we can write up the inverse of $C$, so it is invertible as well.

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Or, one can use the product theorem for determinants and

An $n\times n$ matrix is invertible (i.e. has rank $n$) iff its determinant is nonzero.