Find linear combination of 16 Γ matrices

[qaok]

Homework Statement

Any spinor matrix can be expressed in a set of 16 linearly independent matrices. In the lecture the 16 Γ_J matrices (J=1 to 16) given are I, γ^0,1,2,3, σ^μυ, (γ^μ)(γ_5), iγ_5. I was asked to express
M = (σ_μυ)(γ_5), (σ_μυ)(σ^μυ), (γ^α)(σ_μυ)(γ_α)
in terms of the 16 given Γ matrices.

Homework Equations

The coefficients of the linear combination of the 16 matrices is
c_J = (1/4) Tr (M (inverse of Γ_J))

The Attempt at a Solution

I'd been trying to find the explicit forms σ^μυ but did not find any table. What I found is that the 16 linearly independent matrices can also be the products of γ^0,1,2,3. But I think I was not supposed to find σ^μυ in terms of products of γ^0,1,2,3 then get the inverse of σ^μυ explicitly.

I'm really bad in linear algebra and have no idea about the inverses of the 16 matrices and the properties of the traces of their inverses. Can anyone please help me? Thank you.

I did not use LaTeX because it did not look right when previewing. Sorry about that.

 

[qaok]

I didn't know that (Γ_J)^2 = ±I. So now the inverse of Γ_J can be found.