Write the following regarding solutions of a system
→ x′= A → x: 1. The deï¬nition of a Fundamental Matrix Φ(t), andapply it in one example of your choosing (for a constant matrix Athat you pick). 2. On your example in part 1. calculate thefunction y(t) = detΦ(t) (the determinant of Φ(t) and check thaty(t) solves the ï¬rst order ODE y′(t) = (trA)y(t) (the trace (trA)is the sum of the entries on the main diagonal of A). 3. Choose aconstant 2×2 matrix A different from the one used before. Bysolving the ODE y′(t) = (trA)y(t) conclude that y(t) is eitheridentically zero, or is never zero.
