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hi folks this is linear algebra take-home - were asked to recall the trace of a matrix definition for that so what is the definition of this its you take a square matrix and you add up its diagonal entries so thats what the trace is and were asked to show that if a and B are both and by n matrices then the trace of the product a B is the same as the trace of the product B a so even though in matrix multiplication the order matters in other words a B in general is not equal to B a it turns out that if you add up the diagonal entries youll get the same thing okay so this is going to be a computational proof and like so many we do in class lets let just lets let C be the product of a B so whats the IJ entry of C well by definition its the sum from k equal 1 to n a I K B KJ now Im interested in the trace of a B so Im interested in the trace of C the trace of C is the sum from I equal 1 to N all these matrices are M by n of the i entry of C whats the iin tree of C well heres
