Let $A(t)$ be a continuous family of $n \times n$ matrices and let $P(t)$ be the matrix solution to the initial value problem $P'(t)=A(t)P$, where $P(0)=P_0$. Show that $$\det(P(t))=\det(P_0) e^{\int_0^t \operatorname{Tr} A(s)\, ds}.$$
If we consider $A(t)= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$, $P(t)= \begin{pmatrix} x_1 & y_1 \\ x_2 & y_2 \\ \end{pmatrix}$, and $W(t)=\det (P(t))$ then $W'=\operatorname {Tr}(A) W$.
My question is how do I solve the last system?
And I just consider $2\times 2$ matrix, but the result follows easily for an $n\times n$ matrix; am I right?