This problem is a subset of master course examinations .
I am little good at linear algebra of transfer examinations however I had never saw this problem during studying for 3rd year transfer examinations .
$$ X:= \pmatrix{5&3\\2&6} $$
As $~ Y^{n}=X ~$ is held for $~ n \in \mathbb{N} ~$ , Find out $~ Y ~~ \leftarrow~~ \text{This statemant is as same as the problem statement } ~$
I rewrote the above to the down one . I think both are completely same .
Find out $~ Y \ni Y^{n}=X ~~ \leftarrow~~ n \in\mathbb{N} ~$
Which website(s) for formula(s) should I refer , to solve this problem?
If you know how to diagonalise matrices, you will get using standard methods: $$X=P\,D\,P^{-1}$$ with $$P = \begin{bmatrix} 1 & -3/2 \\ 1 & 1 \end{bmatrix}$$ and $$D = \begin{bmatrix} 8 & 0 \\ 0 & 3 \end{bmatrix}$$ It is not hard to show that for any integer $n \geq 1$ and any matrix $A$: $$(P\,A\,P^{-1})^n = P\,A^n\,P^{-1}$$ So if one takes: $$A = \begin{bmatrix} 8^{1/n} & 0 \\ 0 & 3^{1/n} \end{bmatrix}$$ one gets $A^n = D$ so: $$ (P\,A\,P^{-1})^n = P\,A^n\,P^{-1} = P\,D\,P^{-1} = X$$ So one possible solution is $$Y = P\,A\,P^{-1}$$