Let T be a nilpotent linear operator on the vector space $\mathbb{R}^5$. Let $d_i$ denote the dimension of the kernel $T^i$.Which of the following can possibly occur as a value of $(d_1,d_2,d_3)$?
(a)$(1,2,3)$
(b)$(2,3,5)$
(c)$(2,2,4)$
(d)$(2,4,5)$
I have tried this problem to solve using the inequality $$rank(A)+rank(B)-n\le r(AB) \leq \min\{rank(A),rank(B)\},$$where A,B are matrices of order n. I found all options are correct. It does not match the answer. Please help. I studied a similar question Nilpotent matrix and relation between its powers and dimension of kernels. When I applied the inequality here it matches answer.
Edit: folks were thinking about this in terms of Jordan Normal Form, so I'll try and edit that in too.
In the answer of the question you cited, the lemma tells you that (b) and (c) cannot be correct since the sequences $(d_{i+1}-d_i)$ are not decreasing.
The first sequence $(1,2,3)$ can occur. For instance take $T$ to be represented by a matrix with zeros everyone except one diagonal above its main diagonal, where it is ones. Each time you take a power of $T$, the diagonal "moves up" and you gain one dimension in your kernel. The matrix that I'm describing here is the single Jordan block matrix $J_5(0)$.
The other sequence $(2,4,5)$ can also occur. I'll write out this example fully because it's harder to describe:
$$ \begin{bmatrix}0 & 1 & 0 & 0 & 0\\ 0 &0& 1&0&0\\ 0&0&0&0&0\\ 0&0&0&0&1\\ 0&0&0&0&0\end{bmatrix}^2 = \begin{bmatrix}0&0&1&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix} $$
In the language of JNF, this matrix is $J_3(0)\oplus J_2(0)$.