Suppose a 7 × 7 block diagonal complex matrix A has blocks
(0), (1), \(\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)\), and \(\left(\begin{array}{ccc}2 \pi i & 1 & 0 \\ 0 & 2 \pi i & 0 \\ 0 & 0 & 2 \pi i\end{array}\right)\) along the diagonal.
Which of the following statements are true?
1
The characteristic polynomial of A is x3(x - 1)(x - 2 πi)3.
2
The minimal polynomial of A is x2(x - 1)(x - 2 πi)3.
3
The dimensions of the eigenspaces for 0, 1, 2 πi are 2, 1, 3 respectively.
4
The dimensions of the eigenspaces for 0, 1, 2 πi are 2, 1, 2 respectively.