In this paper, a class of MATRICES, namely, Drazin-dagger MATRICES, in which the Drazin inverse andthe Moore-Penrose inverse commute, is introduced. Also, some properties of the generalized inverses of these MATRICES, are investigated. Moreover, some results about the Moore-Penrose inverse, the Drazin inverse and the numerical range of some reciprocal MATRICES are obtained. In particular, the relations between reciprocal MATRICES, Drazin-Dagger MATRICES and star order are established. Also, some properties of the generalized inverses of the conjugate EP MATRICES are studied. To illustrate the results, some numerical examples are also given.

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The relations between the spectrum of the matrix Q+ Rand the spectra of MATRICES (g+δ) Q+(a+b) R-QR-RQ, QR-RQ, aRb-QRQ, aRQR-(QR) 2, and bR-QR have been given assuming that the matrix Q+R is diagonalizable, where Q, R are {g,δ}-quadratic matrix and f; g-quadratic matrix, respectively, of order n.

This paper uses unit lower triangular MATRICES to solve the nonnegative inverse eigenvalue problem  for various sets of real  numbers. This problem  has remained unsolved for many years for $n \geq 5.$  The inverse of the unit lower triangular MATRICES can be easily calculated and the matrix similarities are also helpful to be able to solve this important problem to a considerable extent. It is assumed that in the given set of eigenvalues, the number of positive eigenvalues is less than or equal to the number of nonpositive  eigenvalues to find a nonnegative matrix such that the given set is its spectrum.