Suppose I am given a $3 \times 3$ matrix, and I need to compute its eigenvalues. How would I be able to tell, by inspection, if the matrix has at least one eigenvalue $= 0$?
Then suppose I know that one of eigenvalues are in fact zero. Are there any shortcuts/simplifications that will allow me to compute the other eigenvalues without having to go through the same tedious calculations for the eigenvalues (and eigenvectors) of a $3 \times 3$ matrix?
A matrix has $0$ as an eigenvalue if and only if it has nonempty kernel. So if you can inspect a nonzero vector whose image is the $0$ vector, you're set.
Alternatively, a matrix has $0$ as an eigenvalue if and only if its determinant is $0$. If you have been doing this for a little bit, you might be able to compute determinants quickly in your head.
In this case, there really can't be a method to find the other two eigenvalues much shorter than simply calculating the characteristic polynomial. You know that you'll get a quadratic and you can always use the quadratic formula to immediately give the eigenvalues.
