Construct a nondiagonal 2x2 matrix that is diagonalizable bu

Construct a nondiagonal 2x2 matrix that is diagonalizable but not invertible.

Solution

Consider the matrix with
row 1 = [ 1 1 ], row 2= [ 0 1 ]
This matrix is invertible: you see this because the det is not zero; you also see it because the diagonal entries of a triangular matrix are the eigenvalues and they are not zero, so the eigenvalues are not zero and that implies the matrix is invertible.
This matrix is not diagonalizable. The fast answer is that this is the Jordan form but it\'s not diagonal, so it can\'t be diagonalized. (The 1 in row 1, column 2 indicates that there is only one genuine eigenvector, so it can\'t be made into a diagonal matrix.) If you\'re not familiar with Jordan form, then you can work out the eigenvectors: you\'ll discover that there is only one eigenvector, so this 2x2 matrix can\'t be diagonalized. (In order to diagonalize a matrix, you need a full set of eigenvectors: for a 2x2, you need 2 eigenvectors, but this matrix only has one.)