Let P ∈ M4(R) be such that P4 is the zero matrix, but P3 is a nonzero matrix. Then which one of the following is FALSE?
1
For every nonzero vector v ∈ R4, the subset {v, Pv, P2v, P3v} of the real vector space R4 is linearly independent.
2
The rank of Pk is 4 − k for every k ∈ {1,2,3,4}.
3
0 is an eigenvalue of P.
4
If Q ∈ M4(R) is such that Q4 is the zero matrix, but Q3 is a nonzero matrix, then there exists a nonsingular matrix S ∈ M4(R) such that S−1QS = P.