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I 'm new in Stack and I need help with a few questions about linear algebra. I'm trying it but I cannot.

TRUE OR FALSE

1) Let $A,B$ and $C$ be $nxn$ matrices such that $C$ is invertible and $B=C.A.{ C }^{ -1 }$, then ${ A }^{ n }={ C }^{ -1 }.{ B }^{ n }.C$

2) If $A\in M_{ nxn }$ Nonsingular matrix, then $det\left( adj\left( A \right) \right) ={ \left( det\left( A \right) \right) }^{ n-1 }$

3) Let $A,B$ and $C$ be $nxn$ matrices, then ${ \left( A.{ B }^{ -1 }+A.C \right) }^{ t }={ A }^{ t }\left( { \left( { B }^{ t } \right) }^{ -1 }+{ C }^{ t } \right) $

Help Please!

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  • $\begingroup$ No, my first time here $\endgroup$ – Ronny Portillo Oct 22 '14 at 21:28
  • $\begingroup$ @Ronny: He is asking for your views on the problem $\endgroup$ – Swapnil Tripathi Oct 22 '14 at 21:31
  • $\begingroup$ I Tried number 1 using determinants but not get anything $\endgroup$ – Ronny Portillo Oct 22 '14 at 21:36
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Since you're not giving your views on the problem, I'll give you some hints.

1) Can you find an expression for $B^n$ from the given data?

2) Well, you know $A^{-1}=\frac{(adj A)}{|A|}$

3) Try taking $B=I_n$ i.e. identity $n$x$n$ matrix and find a counterexample.

Note: $|A|=det(A)$

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