# Similar matrices-Identity

Can anyone help me to understand what is suppose to do in this question?

"A matrix that is similar to the identity matrix" I should say something about this but I do not understand what is meant to do.

I know that two n x n matrices A and B are similar if $B=P^{-1}AP$.

They also have some properties such as:

The same:

Rank; Characteristic equation; Determinant; Trace; Eigenvalues etc.

However I do not understand the question and consequently I do not know how to start.

Can anyone help me on this?

Thanks

• What is the question, exactly? ("A matrix that is similar to the identity matrix" isn't a question . . .) – Noah Schweber Feb 18 '17 at 18:07
• You mean, you're asked to calculate $P^{-1} I P$, where $I$ is the identity matrix? – Andreas Caranti Feb 18 '17 at 18:11
• it only says. to say something about a matrix that is similar to the identity matrix. I really do not understand what should I do. – user290335 Feb 18 '17 at 18:42

A matrix $A \in M_n(\mathbb{F})$ that is similar to the identify matrix $I_n$ must be in fact the identify matrix (that is, $A = I_n$). To see why, note that if $A$ is similar to $I_n$ then we can find an invertible $P$ such that $$A = P^{-1} I_n P = (P^{-1} I_n) P = P^{-1} P = I_n.$$