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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!

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    How is this well researched? C'mon guys2015-08-03

2 Answers 2

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Of course: $B$ invertible implies $B^T$ invertible, and the product of two invertible matrices is clearly invertible.

This is easily seen from these equations: $BB^{-1}=I\implies (BB^{-1})^T=I\implies (B^{-1})^TB^T=1,$ and the fact that if $X$ and $Y$ are invertible, $(XY)^{-1}=Y^{-1}X^{-1}$.


Perhaps the general properties you should take away are these:

$(XY)^T=Y^TX^T$ and $(XY)^{-1}=Y^{-1}X^{-1}$.

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    @nacnudus of course, a reasonable person can tell this is a typo of the form of an omitted word from context. Than$k$s for indirectly alerting me to it.2015-08-03
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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. $