Let $A$ and $B$ be two $2 \times 2$ matrices with integer entries. Prove that $\det(A+xB)$ is an integer polynomial of the form $$P(x) = \det(A+xB) = \det(B)x^2+mx+\det(A).$$
I tried expanding the determinant of $\det(A+xB)$ for two arbitrary matrices, but it got computational. Is there another way?
Best Answer
The constant term $\mathrm{det}(A)$ comes from setting $x = 0.$ The coefficient $\mathrm{det}(B)$ is the constant term of $x^2 P(1/x) = \mathrm{det}(xA + B),$ again setting $x = 0.$