Each n-by-n matrix with n>1 is a sum of 5 coninvolutory matrices

An \(n\times n\) complex matrix \(A\) is called coninvolutory if \(\bar AA=I_n\) and skew-coninvolutory if \(\bar AA=-I_n\) (which implies that \(n\) is even). We prove that each matrix of size \(n\times n\) with \(n>1\) is a sum of 5 coninvolutory matrices and each matrix of size \(2m\times 2m\) is a sum of 5 skew-coninvolutory matrices. We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix \(M\) is called condiagonalizable if \(M=\bar S^{-1}DS\) in which \(S\) is nonsingular and \(D\) is diagonal.