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A Group in which the elements are Square Matrices, the group multiplication law is Matrix Multiplication, and the group inverse is simply the Matrix Inverse. Every matrix group is equivalent to a unitary matrix group (Lomont 1987, pp. 47-48).
See also Maschke's Theorem
References
Lomont, J. S. ``Matrix Groups.'' §3.1 in Applications of Finite Groups. New York: Dover, pp. 46-52, 1987.