First up is from the movie the Matrix, maybe a little too obvious. (Try to forget the sequels.) The point is that you should do it the matrix way and not try to hold onto your system of equations methods that will limit you. Matrices seem so boring on the face of it, but actually they are more powerful than you think.
Second we have our software, something that is willing to do all the work. During tests it will be you alone, but we have a real work hourse in Scilab to do the work. Sam and Dave provide two hits to highlight the tool. Soul Man and Hold on I'm comin'.
The last ones are names of techniques. The main tool is rref. It is always at the start of the problem. Once you have done the rref, there is more to do. This is like the James Bond theme, when you hear it, you know the action is going to start shortly.
The determinate is a love/hate technique. It seems helpful, but it is evil. What better than the third man theme from the movie of the same name. Watch out the determinate can lead you down into the sewers of Vienna.
Set me free, is a line from the tune I've picked for finding eigenvectors. It assumes we already know the eigenvalue, already did the rref. This is because we get to pick the values of the pivot FREE variables.
Finally the whole eigenvalue problem. If you do this wrong the determinate will zither you. But if you try you will get what you need.