Suppose A is a square matrix and v is an eigenvector of A with associated eigenvalue 23. If x is a real number, compute A(xv) in terms of x and v.

Question

Suppose A is a square matrix and v is an eigenvector of A with associated eigenvalue 23. If x is a real number, compute A(xv) in terms of x and v.

A(xv) = (?)

Not sure where to go with this question, is there a rule or condition that I am missing?

So far I thought it would be something using the following;

det(A - xI) = 0 and Ax = vx and solving for A but any simplification using these ideas have been wrong so far. Any suggestions?

Answer 1

If $v$ is an eigenvector with eigenvalue $23$, this means $Av=23v$. So for any real number $x$, we have

$$A(xv)=xAv=x\cdot23 v=23(xv).$$