# Similar matrices and main diagonal summation?

zjohnson19

## Homework Statement

True or False? If A is an n × n matrix, P is an n × n invertible matrix, and B = P −1AP, then
a11 + a22 + . . . + ann = b11 + b22 + . . . + bn

## Homework Equations

Diagnolization, similar matrixes

## The Attempt at a Solution

the question is asking if the summation of the main diagnols of A and B are the same. B is known to be similar to A since B = P^-1 AP. I can't find a counterexample, so I am assuming the summation of the diagnols of both are in fact equal, but this is hardly a proof. I know A and B share the same determinant, rank, and eigenvalues and rank, but I'm not sure how these relate to the main diagnol of the matrices.

Homework Helper
Are you familiar with the concept of trace of a matrix?

Last edited:
$$\textrm{Tr }ABC = \sum_{i=1}^n \sum_{j=1}^n (AB)_{ij}C_{ji}$$