Matrix Differential Equation $P'(t)= A(t)P(t)$

Question

Given $n\times n$ matrix $P(t)$ and $A(t)$ , if $P(t)$ satisfies the matrix differential equation $P'(t)= A(t)P(t)$ and the initial condition $P(0)=P_0$. Then Prove $$\det P(t) = \det P_0 \times \exp\left(\int_0^t tr(A)(s)\,ds\right)$$

If the matrix $A$ is constant matrix then it is easy. But I don't know how to prove it when $A = A(t)$ which has dependent on variable $t$. Please help...

Answer 1

What you are looking for is the Liouville's formula. (See also the proof in the linked article.)