# That Makes It Invertible!

There are several ways of determining whether an $n \times n$ matrix ${\bf A}$ has an inverse:

1. $\det {\bf A} \ne 0$
2. The span of the row vectors is $\mathbb{R}^n$
3. Every matrix equation ${\bf Ax} = {\bf b}$ has a unique solution
4. The row vectors are linearly independent
5. When applying Gaussian elimination, ${\bf A}$ reduces to the identity matrix ${\bf I}$
6. The only solution of ${\bf Ax} = {\bf 0}$ is the trivial solution ${\bf x} = {\bf 0}$
7. ${\bf A}$ has only nonzero eigenvalues
8. The rank of ${\bf A}$ is equal to $n$

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.