Q:

Facts About Eigenvectors and Eigenvalues

I am currently studying Eigenvalues and Eigenvectors in college, and I am supposed to mark true or false on a list of facts about them. I have done so, but it is saying that I am incorrect on the ones that I have marked; I believe that the quiz may be broken, so I was hoping to get some clarification. The facts are:

If Ax=λx for some vector x, then λ is an eigenvalue of A.
Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy.
A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution x.
A matrix A is not invertible if and only if 0 is an eigenvalue of A.
To find the eigenvalues of A, reduce A to echelon form.
If Ax=λx for some vector x, then x is an eigenvector of A.
An eigenspace of A is just a null space of a certain matrix.
The eigenvalues of a matrix are on its main diagonal.
A steady-state vector for a stochastic matrix is actually an eigenvector.
If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.

I believe that the facts that are true are 1, 3, 4, 5, 6, 7, 8, 9. Would someone be able to clarify which are true and false? Thank you.

A:

False, because $x$ might be zero.
True. Finding eigenvectors requires solving a linear system, while the verification just requires a matrix-vector multiplication.
True, this is essentially the definition of an eigenvalue.
False, $A$ is invertible if and only if 0 is not an eigenvalue of $A$.
False, row operations can change the eigenvalues of a matrix.
False. Again, $x$ might be zero.
True, it's the null space of $A - \lambda I$, where $\lambda$ is an eigenvalue.
False, this holds for triangular matrices, but not in general.
True. A steady-state satisfies $Ax = x$, so $x$ is an eigenvector corresponding to $\lambda = 1$.  
False. Consider, for example, the identity matrix. Both $(0,1)$ and $(1,0)$ are eigenvectors corresponding to $\lambda = 1$.

