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Suppose $A$ is a $4\times4$ matrix with $\det A=2$. Find $\det((1/2) A^T A^7 I A^T A^{-1})$ where $I$ is a $4\times4$ identity matrix.

My work so far:

We know that $\det A^T=\det A$.

$I$ has no effect on the determinant.

$det A^{-1}$ is $1/\det A$.

With that said, I think it looks a little like this? $(1/2)\det(A^8)$. (Is it possible to take out the scalar?)

I strongly believe this is not the answer, though.

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No. When you take a scalar $\lambda$ out, it yields $\lambda^4$. Think about the determinant of $\lambda I_4$. –  1015 Mar 6 '13 at 2:33

2 Answers 2

up vote 3 down vote accepted

Be careful about the $1/2$ factor - the determinant is multilinear in the rows (and columns), so if you want to pull a constant out, you must pull it out from each row (or each column). I'll leave that as a hint for how to deal with the scalar - leave a comment if you are still stuck and I'll give more explicit details.

Also, you are correct that $\det(A^{-1}) = 1/\det(A)$, but what you wrote is not correct as written. (It's important to be precise when writing mathematics.)

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Would the answer be 16? Since it is a 4x4 matrix we will have to pull out four 1/2 terms. Multiplying these half terms by 2^8 (128) yields 16. –  EngGenie Mar 6 '13 at 2:34
    
@EngGenie: $2^8$ is $256$. –  Javier Badia Mar 6 '13 at 2:42
    
Yes, sorry. The final answer used 256 as a factor, though! So it is still 16. –  EngGenie Mar 6 '13 at 2:44
    
Yep, that's it. :) –  Michael Joyce Mar 6 '13 at 2:56

Identity to use: $\det(AB)=\det(A).\det(B)$ , $\det(A^T)=\det(A)$ , $\det(A^{-1})=1/\det(A)$

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