What should I do next to show such $A$, $B$ of size $2\times2$ that for all $t \in \Bbb R$ the matrix $A+tB$ has eigenvalues $\lambda_{1,2} (t) = \pm \sqrt t$?
At the moment, I got this: $$tr(A+tB)=\sqrt t - \sqrt t$$ $$tr(A)+t \cdot tr(B)=0$$ $$(a_{11}+a_{22})+t(b_{11}+b_{22})=0$$