Prove that dirac matrices have a vanishing trace

[elduderino]

Not a Homework problem, but I think it belongs here.

1. The problem statement, all variables and given/known data
Consider four dirac matrices that obey

M_i M_j + M_j M_i = 2 \delta_{ij} I

knowing the property that Tr ABC = Tr CAB = Tr BCA show that the matrices are traceless.


2. Relevant equations

Tr MN = Tr NM

3. The attempt at a solution

The square of each dirac matrix is a unit matrix according to the definition above. For i,j unequal

M_i M_j = - M_j M_i

Since these matrices are equal their traces should be equal

Tr M_i M_j = - Tr M_j M_i =- Tr M_i M_j

implying Tr M_i M_j = Tr M_j M_i = 0 for i \neq j

So far, I have not been able to prove that each of these dirac matrices individually as a vanishing trace. I tried

Tr M_i M_j = \sum_k \sum_r (M_i)_{kr}(M_j)_{rk} = 0

but cant conclude anything. This is embarrasing, as this seems pretty basic. Can someone help?

[CompuChip]

Try using the cyclic trace property to show that
\operatorname{Tr}(M_i M_j) = \frac12 \operatorname{Tr}\left( M_i M_j + M_j M_i \right)