If u and v are both eigenvectors of an n ×n matrix A, then u+v is also an eigenvector of the A.

Question

Quizplus · Accepted Answer

Let's consider a counterexample. 
Suppose A is a diagonal matrix with entries 1 and 2 on the diagonal, and let u = [1,0] and v = [0,1]. Both u and v are eigenvectors of A with eigenvalue 1. However, u+v = [1,1] is not an eigenvector of A, since Au+v = [3,3] which is not a scalar multiple of [1,1]. Therefore, the statement is false.

If U and V Are Both Eigenvectors of an N