Commuting Matrices

**NonCommAlg** · August 15th 2010, 10:44 AM

[ $\star \star$ ] Let $A,B$ be any $2 \times 2$ matrices with entries from $\mathbb{C}$ and $AB=BA.$ Let $t_A$ and $I$ denote the trace of $A$ and the $2 \times 2$ identity matrix respectively.

Find some $\alpha, \beta \in \mathbb{C}$ such that $(2A - t_AI)B=\alpha A + \beta I.$

**NonCommAlg** · August 16th 2010, 06:29 AM

i don't have to mention that i'm not interested in a direct solution. anyway, here is a hint for those who like the problem:

Spoiler:

by the way, the result can be extended to $n \times n$ commuting matrices but the solution to this general case is much harder.

**halbard** · August 20th 2010, 12:49 AM

This is a particular consequence of a very well-known result. For any $2\times2$ matrices $A$ and $B$

$AB+BA=t_B A+t_A B+(t_{AB}-t_At_B)I$

In particular, if $AB=BA$ then

Spoiler: