Algebra and Geometry | Part IA, 2001

Let A,B,CA, B, C be 2×22 \times 2 matrices, real or complex. Define the trace trC\operatorname{tr} C to be the sum of diagonal entries C11+C22C_{11}+C_{22}. Define the commutator [A,B][A, B] to be the difference ABBAA B-B A. Give the definition of the eigenvalues of a 2×22 \times 2 matrix and prove that it can have at most two distinct eigenvalues. Prove that a) tr[A,B]=0\operatorname{tr}[A, B]=0, b) trC\operatorname{tr} C equals the sum of the eigenvalues of CC, c) if all eigenvalues of CC are equal to 0 then C2=0C^{2}=0, d) either [A,B][A, B] is a diagonalisable matrix or the square [A,B]2=0[A, B]^{2}=0, e) [A,B]2=αI[A, B]^{2}=\alpha I where αC\alpha \in \mathbb{C} and II is the unit matrix.

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