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.

In this paper, some elementary operations on triangular fuzzy numbers (TFNs) are defined. We also define some operations on triangular fuzzy MATRICES (TFMs) such as trace and triangular fuzzy determinant (TFD). Using elementary operations, some important properties of TFMs are presented. The concept of adjoints on TFM is discussed and some of their properties are. Some special types of TFMs (e.g. pure and fuzzy triangular, symmetric, pure and fuzzy skew-symmetric, singular, semi-singular, constant) are defined and a number of properties of these TFMs are presented.

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.

In this paper, the concept of k-regular fuzzy matrix as a general- ization of regular matrix is introduced and some basic properties of a k-regular fuzzy matrix are derived. This leads to the characterization of a matrix for which the regularity index and the index are identical. Further the relation between regular, k-regular and regularity of powers of fuzzy MATRICES are dis- cussed.

In this article‎, ‎we study the product of 2×2 MATRICES from both geometric and probabilistic perspectives‎, ‎with a particular focus on the concept of growth‎. ‎By exploring the connections between the multiplicative properties of MATRICES and the composition of isometries of hyperbolic geometry‎, ‎new proofs for several classical theorems are presented.