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

- The span of the row vectors is
- Every matrix equation has a unique solution
- The row vectors are linearly independent
- When applying Gaussian elimination, reduces to the identity matrix
- The only solution of is the trivial solution
- has only nonzero eigenvalues
- 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.