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...Cayley hamilton theorem for jordan form matrices if a is an n matrix the describes explict polynomial t of degree characteristic such that one can dene it using determinants but since axler does not treat these until last chapter in book we have to give ad hoc description here which valid namely hence upper triangular then y j derivation amatrix block sum elementary k c some i xed and all values through ci with remaining cases follows immediately suppose now p q wherep andqaresquarematrices foruppertriangularmatricesofthistypewehave zero by induction similar result holds sums arbitrary nite number summands say b where each r see previous discussion factor on right hand side second equation this yields desired identity final remark be dened square has following key property ap invertible however denition proof its use theory yet been covered course...