For both cases indicate whether there exists a real 4 × 4-matrix A with the given properties. Here I denotes the 4 × 4 identity matrix.
$A^2 =0$ and $A$ has rank $3$
$A$ has rank $2$, and $A − I$ has rank $2$.
What I have tried so far for the first problem is some algorithm that was described in my book but I don't think i'm doing it right. The idea is is to use the fact that a nilpotent matrix is similar to a block diagonal matrix. Here $r_j$ = dim ker $A^j$. Then we compute $s_j$ = $r_j$ - $r_{j-1}$ which is the size of the block and $t_j$ = $s_j$ - $s_{j+1}$ which is the corresponding number of blocks. Doing this gives me $r_0$ = 0, $r_1$ = 1, $r_2$ = 4, $r_3$ = 4. Such that $s_1$ = 1, $s_2$ = 3, $s_3$ = 0. And $t_1$ = -2, $t_2$ = 3. Which seems incorrect because how can a 4x4 Matrix A have -2 blocks of size 1 and 3 blocks of size 3.
The number of blocks for an eigenvalue $\lambda_i$ is given by $dim\ rank = (A-\lambda_i I)$.
You also have to take into account that a nilpotent matrix has all eigenvalues equal to zero. Then you can compute the Jordan block diagonal matrices for both cases and you will be able to check the existence of such matrices.