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In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving

Let

The Moore-Penrose inverse of

The two-sided generalized Sylvester matrix equation

The high research activities on Sylvester-type matrix equations can be observed lately. In particular, we note the following papers concerning methods of their computing solutions. Liao et al. [

Systems of periodic discrete-time coupled Sylvester quaternion matrix equations [

Special solutions to Sylvester-type quaternion matrix equations have been actively studied. Roth’s solvability criteria for some Sylvester-type matrix equations were extended over the quaternion skew field with a fixed involutive automorphism in [

Many authors have paid attention also to the Sylvester-type matrix equation involving

Motivated by the vast application of quaternion matrices and the latest interest of Sylvester-type quaternion matrix equations, the main goal of the paper is to derive explicit determinantal representation formulas of the general, Hermitian, and skew-Hermitian solutions to (

Determinantal representation of a solution gives a direct method of its finding analogous to classical Cramer’s rule that has important theoretical and practical significance. However, determinantal representations are not so unambiguous even for generalized inverses within the complex or real settings. Through looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses (see, e.g., [

Other researchers also used the row-column determinants in their developments. In particular, Song derived determinantal representations of the generalized inverse

The paper is organized as follows. In Section

We commence with the following preliminaries which have crucial function in the construction of the chief outcomes of the following sections.

Due to noncommutativity of quaternions, a problem of defining a determinant of matrices with noncommutative entries (which is also defined as noncommutative determinants) has been unsolved for a long time. There are several versions of the definition of noncommutative determinant (see, e.g., [

Suppose

Similarly, for a column determinant along an arbitrary column, we have the following definition.

So an arbitrary

Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in [

Let

For introducing determinantal representations of the Moore-Penrose inverse, the following notations will be used.

Let

Let

Let

If

For an arbitrary full-rank matrix

First note that

Since the projection matrices

If

If

Determinantal representations of orthogonal projectors

Let

Let

Let

Let

Let

Eq. (

In that case, the general solution to (

Some simplifications of (

If

Since

Let

(i)

(ii)

(iii)

(iv)

or

where

(v)

Now consider (

Let

Equation (

In that case, the general solution to (

By putting

Let

(i)

(ii)

(iii)

(iv)

(v)

The proof evidently follows from the proof of Theorem

(i) For the first term of (

If we denote by

(ii) For the second term

(iii) For the third term

(iv) Due to Theorem

(v) Finally, for the second term

Due to [

Suppose that matrices

The general Hermitian solution to (

(i)

(ii)

(iii)

Similarly,

(i)

(ii)

By Lemma

In this section, we give an example to illustrate our results. Let us consider the matrix equation

Now, we find the solution to (

Similarly, we can obtain for all

Within the framework of the theory of row-column determinants, we have derived explicit determinantal representation formulas (analogs of Cramer’s rule) of the general, Hermitian, and skew-Hermitian solutions to the Sylvester-type matrix equation

The data used to support the findings of this study are included within the article titled “Determinantal Representations of General Solutions to the Generalized Sylvester Quaternion Matrix Equation and Its Type”. The prior studies (and datasets) are cited at relevant places within the text.

The author declares that he has no conflicts of interest.