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

Let A and B 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 = Ax. Define the function y(t) = Bx(t). Show that y(t) is also a solution to the name linear system. Consider the non-homogeneous linear system.

Solution

2.

MUltiplying both sides by B gives

Bx\'=BAx

BA=AB

hence

(Bx)\'=A(Bx)

Hence, BX is a solution to :x\'=Ax