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
Best Answer
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 $$A = P^{-1} I_n P = (P^{-1} I_n) P = P^{-1} P = I_n. $$