Matrix $A$ is such that $A^{2}=2 \mathrm{~A}-\mathrm{I}$. Where $I$ is the identity matrix. Then for

Question

Matrix $A$ is such that $A^{2}=2 \mathrm{~A}-\mathrm{I}$. Where $I$ is the identity matrix. Then for $n \geq 2, \quad A^{n}=\ldots$\

Professor PC Thomas Classes · Answer

(a) $\quad A^{2}=2 A-I$\ $A^{3}=2 A^{2}-A$\ $2(2 A-I)-A$\ $3 A-2 I A^{n}=n A-(n-1) I$\