Prove that dirac matrices have a vanishing trace

[elduderino]

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

Homework Statement

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.

Homework Equations

$$Tr MN = Tr NM$$

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 can't conclude anything. This is embarrasing, as this seems pretty basic. Can someone help?

 

[CompuChip]

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