How to construct a upper triangular matrix that satisfy the quadratic polynomial equation with different roots

Abstract

Let $R$ be an associative ring with identity $1$. We describe all matrices in $T_n(R)$ the ring of $n\times n$ upper triangular matrices over $R$ ($n\in \mathbb{N}$), and $T_{\infty}(R)$ the ring of infinite upper triangular matrices over $R$, satisfying the quadratic polynomial equation $x^2-rx+s=0$. For such propose we assume that the above polynomial have two different roots in $R$. Moreover, in the case that $R$ in finite, we compute the number of all matrices to solves the matrix equation $A^2-rA+sI=0,$ where $I$ is the identity matrix.