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Matrices that commute

Source: Romanian District Olympiad 2014, Grade 11, P3

June 15, 2014
linear algebramatrixcomplex numbers

Problem Statement

[*]Let AA be a matrix from M2(C)\mathcal{M}_{2}(\mathbb{C}), A≠aI2A\neq aI_{2}, for any a∈Ca\in\mathbb{C}. Prove that the matrix XX from M2(C)\mathcal{M} _{2}(\mathbb{C}) commutes with AA, that is, AX=XAAX=XA, if and only if there exist two complex numbers α\alpha and α′\alpha^{\prime}, such that X=αA+α′I2X=\alpha A+\alpha^{\prime}I_{2}. [*]Let AA, BB and CC be matrices from M2(C)\mathcal{M}_{2}(\mathbb{C}), such that AB≠BAAB\neq BA, AC=CAAC=CA and BC=CBBC=CB. Prove that CC commutes with all matrices from M2(C)\mathcal{M}_{2}(\mathbb{C}).
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