Comments on: True or False: If $A, B$ are 2 by 2 Matrices such that $(AB)^2=O$, then $(BA)^2=O$ https://yutsumura.com/true-or-false-if-a-b-are-2-by-2-matrices-such-that-ab2o-then-ba2o/ Wed, 16 Aug 2017 19:55:44 +0000 hourly 1 https://wordpress.org/?v=5.3.4 By: Kuldeep Sarma https://yutsumura.com/true-or-false-if-a-b-are-2-by-2-matrices-such-that-ab2o-then-ba2o/#comment-2164 Wed, 16 Aug 2017 19:55:44 +0000 https://yutsumura.com/?p=4639#comment-2164 This problem can also be done in this way. Let C=AB and D=BA. Then both C,D are 2 by 2 matrices and according to given condition C^2=0. Hence C is a nilpotent matrix. So all eigenvalues of C=AB are 0 and we know for square matrices,AB and BA have same set of eigenvalues, So all eigenvalues of BA are zero. Suppose p(t) be the characteristic polynomial of D=BA. Then we have p(t)=t^2. And by cayley hamilton theorem, we have p(D)=0 or D^2=0 or (BA)^2=0.

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