Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 6 / b / Solution

The trace of a square matrix is $\mathrm{tr}A={\sum }_i{A}_{ii}$. Expanding a product and interchanging its finite sums givesHence every commutator has trace zero. Taking the trace of $AB-BA=cI$ gives $0=cn$. Over the complex numbers, with $n\ge 1$, this forcesFor that value, choose $A=B=0$, or any commuting pair. For every nonzero $c$ the trace contradiction proves nonexistence. The finite matrix size is essential to this trace argument.