Let M be 10*10 matrix such that $M^{4}=0$ and $M^{i} \neq 0$ for i=1,2,3 . Then what is the minimum and maximum rank of matrix .

Question

Let $M$ be $10\times10$ matrix such that $M^{4}=0$ and $M^{i} \neq 0$ for $i=1,2,3$. Then what are the minimum and maximum ranks of the matrix. Where minimum rank and maximum rank is r and R respectively ?

a) R= 7, r=3

b)R= 6, r=4

c)R= 6, r=3

d)R= 7, r=4

think since there $M^{i}$ is not equal to $0$ for three values of $i$ so the minimum rank is $3$ But how to find the maximum rank? I have not studied it's not part of curriculum that I have to solve without Jordan form.

Related: https://math.stackexchange.com/questions/863063/upper-bound-for-the-rank-of-a-nilpotent-matrix-if-a2-ne-0. — StubbornAtom, Jan 30 '19 at 16:39

score 2 · Answer 1 · answered Jan 30 '19 at 01:25

2

Your answer for the minimal rank is correct.

I find it easiest to think of this question in terms of the Jordan form of $M$. Because Because $M$ has minimal polynomial $p(x) = x^4$, we know that $M$ has $0$ as its only eigenvalue, and that the largest Jordan block has size $4$. On the other hand, $\dim \ker(A) = 10 - \operatorname{rank}(A)$ is the total number of Jordan blocks associated with zero.

Thus, our question amounts to finding the Jordan form matrix (with only zero-blocks) that contains the fewest blocks, each block of size at most $4$. Since $M$ must contain at least $10/4 = 2.5$ such blocks, we conclude that the kernel has dimension at least $3$, which is to say that the rank of $M$ is at most $7$.

answered Jan 30 '19 at 01:25

Ben Grossmann

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I have not studied it's not part of curriculum that I have to solve without Jordan form . – sejy Jan 30 '19 at 02:10
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@sejy I think you could also make a proof based on Sylvester's rank inequality, that is $$ \operatorname{rank}(AB) \geq \operatorname{rank}(A) + \operatorname{rank}(B) - n $$ it would be easier to give you a helpful answer, though, if you edited your question to include more context – Ben Grossmann Jan 30 '19 at 02:19
Now I have written complete question – sejy Jan 30 '19 at 03:33

Let M be 10*10 matrix such that $M^{4}=0$ and $M^{i} \neq 0$ for i=1,2,3 . Then what is the minimum and maximum rank of matrix .

1 Answers1