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So, the question is to find $\det(X)$ if $2X^T = A^{-1}BA$, where $T$ indicates a transpose, given that $A,B,X$ are $4\times 4$ matrices where $\det(A) = 4$ and $\det(B^{-1}) = 3$. How would I go about doing this?

Siong Thye Goh
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anon.ymous
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  • How would you go about doing this? Step by step, but with some work of yourself. Take the left hand side first. How can we get rid of $2$. Then search for $\det(X^T)$ in terms of $\det (X)$. – Dietrich Burde Nov 01 '17 at 08:58
  • sorry, I wasn't clear at all. The main gripe i have with this question is that it is giving me an equation to find X, and not $\det(X)$. In the question, I do not see $\det(X)$ anywhere so I'm stuck at that part.

    To be more specific - How would I convert an expression of $X = ABA$ if there are no determinants present?

     – anon.ymous Nov 01 '17 at 09:39
  • You would convert $X=ABA$ by seeing the determinant in the question "So, the question is to find $\det(X)$ if ...". Then the next step would be to think about applying $\det$ to the equation $X=ABA$. – Dietrich Burde Nov 01 '17 at 13:14

1 Answers1

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Hint:

Start by taking determinant on both sides.

Properties that might be helpful:

  • $\det(AB)= \det(A)\det(B)$
  • $\det(A^{-1})=\frac1{\det(A)}$
  • $\det(X)=\det(X^T)$
  • $\det(cX)=c^n\det(X)$.
Siong Thye Goh
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