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I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me alive !

My attempt : I've tried to determine a pattern by induction for A and B to the (2k+1) , but got no result . I've also tried breaking both matrices into sums of identity matrix and another complementary matrix , so I could use the binomial theorem but got nothing once more .

Can someone please find a solution ? I'd be really thankful , thanks a lot !

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Hint

You have that $A^2=\left(\begin{matrix}5/6 & 0 \\ 0 & 5/6\end{matrix}\right)=\frac{5}{6}I,$ where $I$ denotes the identity. So $$A^{2k+1}=A^{2k}A=(A^2)^kA=\left(\frac{5}{6}\right)^kA.$$

By induction, you can show that $$B^n=\left(\begin{matrix} 2^{1-n} & 2^{2-n} \\ -2^{-n-1} & -2^{-n} \end{matrix}\right).$$ In particular,

$$B^{2k+1}=\left(\begin{matrix} 2^{-2k} & 2^{1-2k} \\ -2^{-2k-2} & -2^{-2k-1} \end{matrix}\right).$$

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