The commutator of two matrices

Question

The commutator [X, Y] of two matrices is defined by the equation

$$\begin{align} [X, Y] = XY − YX. \end{align}$$

Two anti-commuting matrices A and B satisfy

$$\begin{align} A^2=I \quad B^2=I \quad [A,B]=2iC. \end{align}$$

(a) Prove that $C^2=I$ and that $[B,C] = 2iA$.

I'm thinking how to prove $C^2=I$.

By proving $[A,B]^2=-4I$ we can conclude that $C^2=I$.

\begin{align} (AB - BA)(AB - BA) =\\ = ABAB - ABBA - BAAB + BABA =\\ = ABAB + BABA - 2I \end{align}

Is $ABAB = -I$ ? Is my reasoning correct?

Answer 1

Since $A$ and $B$ anti-commute, then it follows \begin{align} ABAB = -AABB = -I. \end{align}

Answer 2

For sake of completeness I'll show the second part of the question: $$ [B,C]=BC-CB=B(-iAB)-(-iAB)B= \\ =-iBAB+iABB=-iBAB+iA=iABB+iA= \\ =iA+iA=2iA. $$

Where it is used $AB=iC$ since $[A,B]=2AB$.