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If $A$ and $B$ are two invertible 5*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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A common trick to answer the question "Is foo invertible?" is to actually write down the inverse of foo. In many cases this is pretty easy to do, as seen in the answers. –  Hurkyl Oct 24 '12 at 16:10
    
@Hurkyl What does foo mean? –  J.A.F Oct 24 '12 at 16:13
    
@PENGTENG It is a nonsense word used as a generic placeholder in math and computer science. It's being used as a variable for an object here, but it's more frequently used for properties rather than objects. –  rschwieb Oct 24 '12 at 16:14
    
@rschwieb I get it. Thanks! –  J.A.F Oct 24 '12 at 16:17

2 Answers 2

up vote 8 down vote accepted

Of course: $B$ invertible implies $B^T$ invertible, and the product of two 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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And this is of course true for $n \times n$-matrices and not just $5 \times 5$-matrices. –  N.U. Oct 24 '12 at 16:11

Yes. $$ \det(B^T\,A)=\det(B^T)\det(A)=\det(B)\det(A)\ne0. $$ Moreover $$ (B^T\,A)^{-1}=A^{-1}(B^{-1})^T. $$

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