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.