There are several ways of determining whether an matrix has an inverse:

1. 
2. The span of the row vectors is
3. Every matrix equation has a unique solution
4. The row vectors are linearly independent
5. When applying Gaussian elimination, reduces to the identity matrix
6. The only solution of is the trivial solution
7. has only nonzero eigenvalues
8. The rank of is equal to

Of course, it’s far more fun to remember these facts in verse (pun intended). From the YouTube description, here’s a Linear Algebra parody of One Direction’s “What Makes You Beautiful”. Performed 3/8/13 in the final lecture of Math 40: Linear Algebra at Harvey Mudd College, by “The Three Directions.”

While I’m on the topic, here’s a brilliant One Direction mashup featuring the cast of Downton Abbey. Two giants of British entertainment have finally joined forces.