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

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- Solution b

Solution

The trace of a square matrix is

tr A = \sum_{i} A_{ii}

. Expanding a product and interchanging its finite sums gives

tr (A B) = i, j \sum A_{ij} B_{ji} = j, i \sum B_{ji} A_{ij} = tr (B A) .

Hence every commutator has trace zero. Taking the trace of

A B - B A = c I

gives

0 = c n

. Over the complex numbers, with

n \geq 1

, this forces

c = 0.

For 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.

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