<s>
A	O
quaternionic	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
whose	O
elements	O
are	O
quaternions	O
.	O
</s>
<s>
The	O
product	O
of	O
two	O
quaternionic	O
matrices	O
A	O
and	O
B	O
also	O
follows	O
the	O
usual	O
definition	O
for	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
Then	O
the	O
entry	O
in	O
the	O
ith	O
row	O
and	O
jth	O
column	O
of	O
the	O
product	O
is	O
the	O
dot	O
product	O
of	O
the	O
ith	O
row	O
of	O
the	O
first	O
matrix	B-Architecture
with	O
the	O
jth	O
column	O
of	O
the	O
second	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
identity	O
for	O
this	O
multiplication	O
is	O
,	O
as	O
expected	O
,	O
the	O
diagonal	O
matrix	B-Architecture
I	O
=	O
diag( 1	O
,	O
1	O
,	O
...	O
,	O
1	O
)	O
.	O
</s>
<s>
This	O
defines	O
a	O
map	O
Ψmn	O
from	O
the	O
m	O
by	O
n	O
quaternionic	O
matrices	O
to	O
the	O
2m	O
by	O
2n	O
complex	O
matrices	O
by	O
replacing	O
each	O
entry	O
in	O
the	O
quaternionic	B-Algorithm
matrix	I-Algorithm
by	O
its	O
2	O
by	O
2	O
complex	O
representation	O
.	O
</s>
<s>
The	O
complex	O
valued	O
determinant	O
of	O
a	O
square	O
quaternionic	B-Algorithm
matrix	I-Algorithm
A	O
is	O
then	O
defined	O
as	O
det(Ψ(A )	O
)	O
.	O
</s>
<s>
Many	O
of	O
the	O
usual	O
laws	O
for	O
determinants	O
hold	O
;	O
in	O
particular	O
,	O
an	O
n	O
by	O
n	O
matrix	B-Architecture
is	O
invertible	O
if	O
and	O
only	O
if	O
its	O
determinant	O
is	O
nonzero	O
.	O
</s>
