Tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute

This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned matrices to their original forms, we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent.