Question 146
Let $A, B \in \mathcal{M}_{2}(\mathbb{R})$ be matrices such that $(A-B)^{2}=\mathrm{O}_{2}$.
a) Show that $\operatorname{det}\left(A^{2}-B^{2}\right)=(\operatorname{det}(A)-\operatorname{det}(B))^{2}$.
b) Prove that $\operatorname{det}(A B-B A)=0$ if and only if $\operatorname{det}(A)=\operatorname{det}(B)$.
a) Show that $\operatorname{det}\left(A^{2}-B^{2}\right)=(\operatorname{det}(A)-\operatorname{det}(B))^{2}$.
b) Prove that $\operatorname{det}(A B-B A)=0$ if and only if $\operatorname{det}(A)=\operatorname{det}(B)$.