Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix $$A_7\doteq\pmatrix{1&2\cr 9&4\cr}$$ from the Linear Transformations activity since the students have already seen this matrix and know what it's eigenvectors are.

1. We like to start by asking the students what they remember from their math classes about eigenvalues and eigenvectors. Typically our students remember that “It has something to do with a determinant and a $\lambda$,” but not much else.
2. Introduce the eigenvalue equation. If you have used the Linear Transformations activity, then students should know geometrically that the eigenvectors of a transformation are the vectors that are only changed by a scale. If you get the students to say this in words, you can write the eigenvalue equation $A\vert v\rangle =\lambda \vert v\rangle$ on the board immediately afterwards. This will help them see how the algebraic statement is connected to the geometric statement.
3. Demonstrate methods to solve the characteristic equation using your chosen example.
4. Demonstrate methods to find the eigenvectors associated with a given eigenvalue using your chosen example.