How do I get the eigenvalues, x, of a matrix, M, with eigenvectors, v?

To start off, let's write the equation involving these three objects: Mv = xv. Now, looking at the first form of the equation, we don't want any eigenvector to have all its entries be zero, or else the eigenvalue could take any value - and doesn't really make sense! (I would explain how we can see this by showing that the matrix multiplying a vector with all zero entries would give us back the same vector, and likewise any number multiplying a vector with all zero entries does the same, using explicit column vectors.) We don't need to know any more about the eigenvectors than that to get the eigenvalues, for which we rearrange our equation: (M - xI)v = 0. (Here I would explain why we're allowed to use the identity matrix if the idea is unfamiliar to the student.)So, given the eigenvectors need to have at least one entry with a non-zero value, we need the bit in brackets from our second equation to be zero. But it's a matrix, so we say we want the determinant to be zero, which is the same thing. (Here I would explain why this is the case if the idea was new to the student.) Solving |M - xI| = 0 gives us an equation in terms of x, which is likely quadratic for a 2x2 matrix and cubic for a 3x3 matrix. (I would show how to perform the determinant of a matrix explicitly if unfamiliar to the student.) This means we get multiple values for x. Each is a valid eigenvalue, and if we wanted to get the eigenvectors we would now take these back to the first equation we wrote down and solve it for each eigenvalue in turn. (I would give a thorough example of this in lesson.)