6.1. An $n \times n$ matrix $A$ is diagonalizable if and only if $A$ has $n$ linearly independent eigenvectors. Find the characteristic polynomial, eigenvalues, and eigenvectors of each of the following matrices, if there exist.
(1) $\begin{bmatrix} 2 & 0\\0 & 3 \end{bmatrix}$,
(2) $\begin{bmatrix} 1 & 2 & 3\\0 & 2 & 3\\0 & 0 & 3 \end{bmatrix}$,
(3) $\begin{bmatrix} -2 & 0 & 0\\3 & 2 & 3\\4 & -1 & 6 \end{bmatrix}$,
(4) $\begin{bmatrix} 0 & 1 & 0 & 1\\1 & 0 & 1 & 0\\0 & 1 & 0 & 1\\1 & 0 & 1 & 0 \end{bmatrix}$,
(5) $\begin{bmatrix} 1 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1 \end{bmatrix}$,
(6) $\begin{bmatrix} 0 & -5 & 0 & 0\\5 & 0 & 0 & 0\\0 & 0 & 0 & -2\\0 & 0 & 2 & 0 \end{bmatrix}$.
