Linear Algebra Homework: Matrix \Gamma^{\mu}p_{\mu} Rank

[Grieverheart]

Homework Statement

Ok, it seems I need to refresh my linear algebra a bit. In the string theory exams, we had a part about space-time supersymmetry of the superparticle. On of the questions was this:

Argue that the matrix [itex]\Gamma^{\mu}p_{\mu}[/itex] can have at most half maximal rank upon imposing the equations of motion.

(Recall that any matrix can be brought to Jordan normal form, meaning that it is diagonal with at most 1s in places right above the diagonal entries.

Homework Equations

Equations of Motion (the relevant ones):
[itex]\Gamma^{\mu} p_{\mu} \dot{\theta}=0[/itex], where [itex]\dot{\theta}(\tau)[/itex] is an anti-commuting spinor in space-time and has 32 components (we're working in 10 dimensions.

The Attempt at a Solution

I write [itex]\Gamma^{\mu}p_{\mu} \dot{\theta}=0[/itex] as (I use 3x3 for simplicity)

[itex]\begin{pmatrix}
a & 1 & 0\\
0 & b & 1\\
0 & 0 & c
\end{pmatrix}
\begin{pmatrix}
\dot{\theta_0}\\
\dot{\theta_1}\\
\dot{\theta_2}
\end{pmatrix}=0[/itex]

From this I get 3 equations, but I'm not really sure how they affect the rank.

 

[Nate810]

I think it has something to do with the fact that the equations are linearly independent and so the rank of \Gamma^{\mu}p_{\mu} can be at most 3-2=1, but I'm not sure how to express this mathematically. Any help would be appreciated.