$2\times 2$ matrix with $2$ Distinct Eigenvalues is Diagonalizable In a class of mine, my teacher said that any $2\times 2$ matrix with $2$ distinct eigenvalues is diagonalizable. I've been trying to find a proof of this or a theorem to support it but cannot find anything. Is this statement true?

We can use the fact that eigenvectors corresponding to distinct eigenvalues are linearly independent. Now recall that an $n\times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors.  
