Part 2

New Skills

Formulating Maths
Introduction
Hypotheses &C
Implications
And & or
For all & there exists
Dependence

Proof
Introduction
Counter examples
Constructing proofs
Understanding
Experimentation
Precision
Examples
Assumptions
Contradiction
Induction
The End

 

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Example 7. Let A and B be n ×n matrices. Prove that the traces of AB and BA are equal.

Definitions: The reader may not know the definition of the trace of a matrix. It can be found in books on linear algebra. The trace of an n×n matrix A is the sum of the entries on the leading diagonal. If the entry in row i, column j of A is denoted by a_ij, then

tr A = a11 + a22 + ... + ann = n å i = 1  aii.

Hypotheses: The only hypotheses are that A and B are n ×n matrices.

Conclusions: We have to prove that tr AB = tr BA.