Condition for $\det(A^{T}A)=0$ Is it always true that
$\det(A^{T}A)=0$, $\hspace{0.5mm}$ for $A=n \times  m$ matrix with $n<m$?
From some notes I am reading on Regression analysis, and from some trials, it would appear this is true.
It is not a result I have seen, surprisingly.
Can anyone provide a proof?
Thanks.

$A^TA$ is an $m\times m$ matrix and has determinant 0 unless its rank is $m$.
However, the rank is the dimension of the image of $\mathbb R^m$ under the linear transformation defined by the matrix ... and this image is a subset of the image of $\mathbb R^n$ under the transformation defined by $A^T$. Since a linear transformation can never increase the dimension of the space it applies to, the rank is at most $n$, which is, by assumption, less than $m$.
