Let A and be n times n matrices Recall that A and B commute

Let A and be n times n matrices. Recall that A and B commute if AB = BA, that is, the order of multiplication does not matter. Suppose that x(t) is a solution to the linear system x overdot = Ax. Define the function y(t) = Bx(t). Show that y(t) is also a solution to the same linear system.

Solution

Multiplying both sides by B gives

Bx\'=BAx

(Bx)\'=ABx

(Bx)\'=A(Bx)

y\'=Ay

Hence y is a solution of the system.