Let $X$ be a nonzero $n × n$ matrix such that $X^m = 0$ for some $m ∈ N$. Let $b ∈ N$ be the smallest number for which $X^b = 0$. Show that $b≤ n$.
41411
How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$
65162
Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
81818
Prove or disprove : $\det(A^k + B^k) \geq 0$
10236
Show that $a^n \mid b^n$ implies $a \mid b$
56818
Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
128043
$A^n=0$ for $A$ nilpotent without the use of minimal polynomial
59513
For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
131876
Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$
117352
Show $x^n+ax+b=0$ has most two solutions
145980
For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$?
31766
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$
86518
Does $X$ have a lower bound?
19046
Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
122622
Does there exist a matrix $P$ such that $P^n=M$ for a special matrix $M$?
81346
Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.