TITLE: Does the product of two invertible matrix remain invertible?
QUESTION [9 upvotes]: If $A$ and $B$ are two invertible $5 \times 5$ matrices, does $B^{T}A$ remain invertible?
I cannot find out is there any properties of invertible matrix to my question.
Thank you!

REPLY [10 votes]: Yes.
$$
\det(B^T\,A)=\det(B^T)\det(A)=\det(B)\det(A)\ne0.
$$
Moreover
$$
(B^T\,A)^{-1}=A^{-1}(B^{-1})^T.
$$