An important application of determinantal representations of generalized inverses is the Cramer rule for generalized inverse solutions of matrix equations.
T,S] , the Bott-Duffin inverse , the Cramer rule for the solutions of restricted matrix equations , the generalized Stein quaternion matrix equation , and so forth.
Recently, within the framework of a theory of the column and row determinants, Song  has considered the characterization of the W-weighted Drazin inverse over the quaternion skew and presented a Cramer rule of the restricted matrix equation:
i) To derive a Cramer rule (37) we use the point (a) from Lemma 8.
To establish a Cramer rule of (32) we will not use the determinantal representations (28) and (28) for (36) because the corresponding determinantal representations of its solution will be too cumbersome.
To derive a Cramer rule (58), we use the determinantal representation (27) for [A.
Sun, "A Cramer rule for solution of the general restricted matrix equation," Applied Mathematics and Computation, vol.