Question 77
Let $A$ and $B$ be two $3 \times 3$ matrices with real entries, so that $AB^2A = BA^2B$.
a) Prove that there exists $\alpha, \beta \in \mathbb{R}$ so that
$$
\det[(AB)^2 + (BA)^2] = [\det(A)\det(B) - \alpha]^2 + [\det(A)\det(B) - \beta]^2.
$$
a) Prove that there exists $\alpha, \beta \in \mathbb{R}$ so that
$$
\det[(AB)^2 + (BA)^2] = [\det(A)\det(B) - \alpha]^2 + [\det(A)\det(B) - \beta]^2.
$$