Please Login to access more options.

Problem 32 (Finding Idempotent And Nilpotent Elements In A Matrix Ring)

Complete the following:

  1. In the ring $M_2(\mathbb{Z})$, the matrices $\begin{bmatrix}0&0\\0&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&1\end{bmatrix}$, $\begin{bmatrix}1&0\\0&0\end{bmatrix}$, $\begin{bmatrix}0&0\\0&1\end{bmatrix}$, are idempotents. Find two others.
  2. Show that $\begin{bmatrix}0&t\\0&0\end{bmatrix}$ is nilpotent in $M_2(\mathbb{Z})$ and that $\begin{bmatrix}0&t&0\\0&0&t\\0&0&0\end{bmatrix}$ is nilpotent in $M_3(\mathbb{Z})$, where $t\in\mathbb{Z}$.
  3. Let $X=\begin{bmatrix}0&t&0&0&0\\0&0&t&0&0\\0&0&0&t&0\\0&0&0&0&t\\0&0&0&0&0\end{bmatrix}$. Compute $X^k$ for each positive integer $k$. What patterns do you see? Is $X$ a nilpotent element of $M_5(\mathbb{Z})$?

The following pages link to this page.