Consider n x n matrices with real entries. Prove a1,1 ... a1,n a1,1 ... a1,n a1.1 ... a1,n det ak,1+bk,1 ... ak,n+bk,n = det ak,1 ... ak,n +det bk. bk.n ... an,1 ...an,n Lan,1 ... an,n- Lan.1 .., an,n- for all integers n. Let the entries be differentiable functions of t and let [a,1(t) an(t)] [a,1(t), an(t)] = V(t) det ak1(t), ak,n(t); V(t)(k) = = det(t)(t) | Lan,1(t) ... ann(t)] Lan,(t), ann(t)] Prove that the derivative w.r.t. to t is V' (t) = v(t)(k) k=1 Let y,(t) for 1in be solutions to the ODE y(n)(t)+p(t)y(n-1)(t)+...+pn(t)y(t) = and let Prove that W(t) =0 9 y(k) (t) = ()ky(t) y(t) ... Yn (t) = det(t) 1) (t) (k-) Lyi (n-1) (t) (n-1), Yn W'(t)=p(t)W(t)