TITLE: Product with Invertible Matrices Question
QUESTION [0 upvotes]: If $A$ is an $n \times k$ matrix and $B$ is a $k \times l$ matrix (for some finite positive integers), and if $B^TB$ is invertible, and if $B^TA^TAB$ is invertible, does this imply $A^TA$ is also invertible?
Edit: What if we assume $A$ is square?

REPLY [0 votes]: In general this is not true; for instance if $B$ is a vector row of size $1\times n$; then $B^TB$ has rank equal to $1$ and so this not invertible unless $n=1$ and $B$ is the non-zero element.
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