 
Summary: ,1
Introduction
The classical matrix groups are of fundamental impor
tance in many parts of geometry and algebra. Some of
them, like Sp(n), are most conceptually defined as
groups of quaternionic matrices. But, the quaternions
not being commutative, we must reconsider some as
pects of linear algebra. In particular, it is not clear how
to define the determinant of a quaternionic matrix. Over
the years, many people have given different definitions.
In this article I will discuss some of these.
Let us first briefly recall some basic facts about quater
nions. The quaternions were discovered on October 16,
1843 by Sir William Rowan Hamilton. (For more on the
history, I recommend [19t [31L [47L and [48].) They
form a noncommutative, associative algebra over IR:
IHl = {a + ib + jc + kd I a, b, c, d E IRL
where
", P= l = k2
