Determinant, Eigenvalue and Eigenvector

The Problem

Consider the matrix

$A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right).$

1. Compute $\det (A)$.
2. Find the eigenvalues ${\lambda }_1,{\lambda }_2$ of $A$.
3. Find a corresponding eigenvector for each eigenvalue.
4. Diagonalise $A$: write $A=PD{P}^{-1}$ with $D$ diagonal.

This is the canonical 2×2 symmetric eigenproblem. Get this in your bones — every PCA, every spectral clustering, every covariance discussion assumes you can do this in your sleep.