# Flag of subspaces

Let $T$ be a $4\times 4$ real matrix such that $T^4=0$. Let $k_i =\dim \ker T^i$ for $i=1,2,3,4$.

Then, which of the following is not a possibility for the sequence $k_1\leq\ k_2\leq\ k_3\leq\ k_4$ ?

1. $3\leq4\leq4\leq4$

2. $1\leq3\leq4\leq4$

3. $2\leq4\leq4\leq4$

4. $2\leq3\leq4\leq4$

• Nullity of $T$ will imply the number of jordan blocks related to $0$. That's all I could know. I din't quite get that answer. The question was actually asked in a national level test for entrance to Phd in India in $2015$ and the answer was given to be the second option May 1, 2017 at 13:29
Hint: Suppose that $T$ is in Jordan canonical form (without loss of generality). Note that for all $i$, $$\dim \ker T^i - \dim \ker T^{i-1}$$ is the number of Jordan blocks with size at least $i$ (here, $T^0$ is the identity matrix).