Suppose that A is an matrix and is a solution to the system of linear differential equations where is an eigenvector of A with associated eigenvalue
. If A is invertible, then is a solution to the system
.

Question

Suppose that A is an  matrix and  is a solution to the system of linear differential equations  where  is an eigenvector of A with associated eigenvalue  
. If A is invertible, then  is a solution to the system  
.

Quizplus · Accepted Answer

Since 11edaba5_20a7_e212_a975_f56e6cd4de18_TBMC1039_00 is a solution to the system of linear differential equations, we know that A11edaba5_20a7_e212_a975_f56e6cd4de18_TBMC1039_00 = 11edaba5_20a7_e213_a975_3ded0e19e1b9_TBMC1039_00. Multiplying both sides by A^-1, we get A^-111edaba5_20a7_e212_a975_f56e6cd4de18_TBMC1039_00 = A^-111edaba5_20a7_e213_a975_3ded0e19e1b9_TBMC1039_00. Since A is invertible, A^-111edaba5_20a7_e213_a975_3ded0e19e1b9_TBMC1039_00 is a unique solution to the system, so 11edaba5_20a7_e216_a975_9d36b16911fb_TBMC1039_00 is also a solution.

Suppose That a Is an Matrix and Is