Found problems: 6
Let \(A\) be a square matrix with entries in the field \(\mathbb Z / p \mathbb Z\) such that \(A^n - I\) is invertible for every positive integer \(n\). Prove that there exists a positive integer \(m\) such that \(A^m = 0\).
[i](A matrix having entries in the field \(\mathbb Z / p \mathbb Z\) means that two matrices are considered the same if each pair of corresponding entries differ by a multiple of \(p\).)[/i]
[i]Proposed by Tony Wang[/i]
For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent.
(i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$.
(ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)
Let $n$ be a natural number $n \geq 2$ and matrices $A,B \in M_{n}(\mathbb{C}),$ with property $A^2 B = A.$
a) Prove that $(AB - BA)^2 = O_{n}.$
b) Show that for all natural number $k$, $k \leq \frac{n}{2}$ there exist matrices $A,B \in M_{n}(\mathbb{C})$ with property stated in the problem such that $rank(AB - BA) = k.$
Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that
\[A^2B-2ABA+BA^2=0.\]
Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.
Let $A$ and $B$ be real $n\times n $ matrices. Assume there exist $n+1$ different real numbers $t_{1},t_{2},\dots,t_{n+1}$ such that the matrices
$$C_{i}=A+t_{i}B, \,\, i=1,2,\dots,n+1$$
are nilpotent. Show that both $A$ and $B$ are nilpotent.
[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring.
[b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.