Find all similar matrices to (alpha)I

Question

Well the question is in the header - find all the similar matrices to skalar * I. I just can't see what i am supposed to do and i didn't find a similar question over the internet so i thought i will ask here maybe someone know the method to solve this type of question.

Answer 1

Two matrices $A,B\in k^{n\times n}$ ( $n\in\mathbb{N}$ and say $k$ a field) are similar if there exists an invertible matrix $P\in k^{n\times n}$ such that $A=P^{-1}BP$. Now for $\alpha\in k$ and $I$ the identity matrix this means every similar matrix is of the form: $$P^{-1}\alpha I P\overset{*}=\alpha P^{-1}IP=\alpha P^{-1}P=\alpha I,$$ where $*$ holds by the compatibilty with scalars rule, which holds since $k^{n\times n}$ is an algebra over $k$.