Is the determinant of a matrix multiplied by its transpose equal to the determinant of the matrix's transpose multiplied by the matrix?

Question

Is

$$ det(AA^T) = det (A^TA) $$

for any rectangular matrix true?

Answer 1

Consider $A=[1\cdots 1]\in M_{1,n}(\mathbb{R})$ and $n\ge2$.

We have $A^TA=n$ and so $\det(A^TA)=n$

But $AA^T$ has rank $1<n$ hence $\det(AA^T)=0$