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2x2 matrices

Source: Romanian District Olympiad 2015, Grade XI, Problem 2

September 26, 2018
linear algebraMatrices

Problem Statement

Let be two matrices A,B∈M2(R) A,B\in M_2\left(\mathbb{R}\right) that satisfy the equality (A−B)2=O2. \left( A-B\right)^2 =O_2.
a) Show that det⁡(A2−B2)=(det⁡A−det⁡B)2. \det\left( A^2-B^2\right) =\left( \det A -\det B\right)^2. b) Demonstrate that det⁡(AB−BA)=0  ⟺  det⁡A=det⁡B. \det\left( AB-BA\right) =0\iff \det A=\det B.
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