<s>
In	O
mathematics	O
,	O
more	O
specifically	O
in	O
multilinear	O
algebra	O
,	O
an	O
alternating	B-Algorithm
multilinear	I-Algorithm
map	I-Algorithm
is	O
a	O
multilinear	O
map	O
with	O
all	O
arguments	O
belonging	O
to	O
the	O
same	O
vector	O
space	O
(	O
for	O
example	O
,	O
a	O
bilinear	O
form	O
or	O
a	O
multilinear	O
form	O
)	O
that	O
is	O
zero	O
whenever	O
any	O
pair	O
of	O
arguments	O
is	O
equal	O
.	O
</s>
<s>
The	O
notion	O
of	O
alternatization	B-Algorithm
(	O
or	O
alternatisation	O
)	O
is	O
used	O
to	O
derive	O
an	O
alternating	B-Algorithm
multilinear	I-Algorithm
map	I-Algorithm
from	O
any	O
multilinear	O
map	O
with	O
all	O
arguments	O
belonging	O
to	O
the	O
same	O
space	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
matrix	O
is	O
a	O
multilinear	O
alternating	B-Algorithm
map	I-Algorithm
of	O
the	O
rows	O
or	O
columns	O
of	O
the	O
matrix	O
.	O
</s>
<s>
If	O
any	O
component	O
of	O
an	O
alternating	B-Algorithm
multilinear	I-Algorithm
map	I-Algorithm
is	O
replaced	O
by	O
for	O
any	O
and	O
in	O
the	O
base	O
ring	O
then	O
the	O
value	O
of	O
that	O
map	O
is	O
not	O
changed	O
.	O
</s>
<s>
The	O
alternatization	B-Algorithm
of	O
an	O
n-multilinear	O
alternating	B-Algorithm
map	I-Algorithm
is	O
n	O
!	O
</s>
<s>
The	O
alternatization	B-Algorithm
of	O
a	O
symmetric	O
map	O
is	O
zero	O
.	O
</s>
<s>
The	O
alternatization	B-Algorithm
of	O
a	O
bilinear	O
map	O
is	O
bilinear	O
.	O
</s>
<s>
Most	O
notably	O
,	O
the	O
alternatization	B-Algorithm
of	O
any	O
cocycle	O
is	O
bilinear	O
.	O
</s>
