On Idempotency of Linear Combinations of a Quadratic or a Cubic Matrix and an Arbitrary Matrix

Abstract

Let ${\bf{A}}$ be a quadratic or a cubic $n \times n$ nonzero matrix and ${\bf{B}}$ be an arbitrary $n \times n$ nonzero matrix. In this study, we have established necessary and sufficient conditions for the idempotency of the linear combinations of the form $a{\bf{A}} + b{\bf{B}}$, under the some certain conditions imposed on $\bf{A}$  and $\bf{B}$, where $a,b$ are nonzero complex numbers.