Thread: Very simple true or false

If λ is an eigenvalue of a matrix A, then there exist infinitely many vectors x such
that Ax = λx. True or False?
I'm guessing that it is true since we will have a general solution where any number can be placed in the variable.
Thanks

Originally Posted by farmeruser1

If λ is an eigenvalue of a matrix A, then there exist infinitely many vectors x such
that Ax = λx. True or False?
I'm guessing that it is true since we will have a general solution where any number can be placed in the variable.
Thanks

The definition of eigenvalue tells you that there must be at least one eogenvector x. But if x is an eigenvector then so is any nonzero scalar multiple of x. So the answer is going to depend on how many elements there are in the field of scalars.