On the equality between rank and trace of an idempotent matrix

The paper was inspired by the question whether it is possible to derive the equality between the rank and trace of an idempotent matrix by using only the idempotency property, without referring to any further features of the matrix. It is shown that such a proof can be obtained by exploiting a general characteristic of the rank of any matrix. An original proof of this characteristic is provided, which utilizes a formula for the Moore–Penrose inverse of a partitioned matrix. Further consequences of the rank property are discussed, in particular, several additional facts are established with considerably simpler proofs than those available. Moreover, a collection of new results referring to the coincidence between rank and trace of an idempotent matrix are derived as well.