Matrix A such that A² =2A - I where I is the identity matrix, then for $\mathrm{n}\ge 2$, Aⁿ is equal to

Given A² =2A - I

Now 

$\begin{array}{l}{\mathrm{A}}^3=\mathrm{A}\left({\mathrm{A}}^2\right)=\mathrm{A}(2\mathrm{A}-\mathrm{I})=2{\mathrm{A}}^2-\mathrm{A}=2(2\mathrm{A}-\mathrm{I})-\mathrm{A}\\ =3\mathrm{A}-2\mathrm{I}\\ {\mathrm{A}}^4=\mathrm{A}\left({\mathrm{A}}^3\right)=\mathrm{A}(3\mathrm{A}-2\mathrm{I})=3{\mathrm{A}}^2-2\mathrm{A}=3(2\mathrm{A}-\mathrm{I})-2\mathrm{A}\\ =4\mathrm{A}-3\mathrm{I}\end{array}$

Following this, we can say ${\mathrm{A}}^{\mathrm{n}}=\mathrm{nA}-(\mathrm{n}-1)\mathrm{I}$