<s>
In	O
mathematics	O
,	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
where	O
all	O
the	O
entries	O
are	O
zero	O
except	O
those	O
on	O
the	O
diagonal	O
going	O
from	O
the	O
lower	O
left	O
corner	O
to	O
the	O
upper	O
right	O
corner	O
(	O
↗	O
)	O
,	O
known	O
as	O
the	O
anti-diagonal	B-Algorithm
(	O
sometimes	O
Harrison	O
diagonal	O
,	O
secondary	O
diagonal	O
,	O
trailing	O
diagonal	O
,	O
minor	B-Algorithm
diagonal	I-Algorithm
,	O
or	O
bad	O
diagonal	O
)	O
.	O
</s>
<s>
All	O
anti-diagonal	B-Algorithm
matrices	O
are	O
also	O
persymmetric	B-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
two	O
anti-diagonal	B-Algorithm
matrices	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
the	O
product	O
of	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
with	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
is	O
anti-diagonal	B-Algorithm
,	O
as	O
is	O
the	O
product	O
of	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
is	O
invertible	O
if	O
and	O
only	O
if	O
the	O
entries	O
on	O
the	O
diagonal	O
from	O
the	O
lower	O
left	O
corner	O
to	O
the	O
upper	O
right	O
corner	O
are	O
nonzero	O
.	O
</s>
<s>
The	O
inverse	O
of	O
any	O
invertible	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
is	O
also	O
anti-diagonal	B-Algorithm
,	O
as	O
can	O
be	O
seen	O
from	O
the	O
paragraph	O
above	O
.	O
</s>
<s>
The	O
determinant	O
of	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
has	O
absolute	O
value	O
given	O
by	O
the	O
product	O
of	O
the	O
entries	O
on	O
the	O
diagonal	O
from	O
the	O
lower	O
left	O
corner	O
to	O
the	O
upper	O
right	O
corner	O
.	O
</s>
<s>
However	O
,	O
the	O
sign	O
of	O
this	O
determinant	O
will	O
vary	O
because	O
the	O
one	O
nonzero	O
signed	O
elementary	O
product	O
from	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
will	O
have	O
a	O
different	O
sign	O
depending	O
on	O
whether	O
the	O
permutation	B-Algorithm
related	O
to	O
it	O
is	O
odd	O
or	O
even	O
:	O
</s>
<s>
More	O
precisely	O
,	O
the	O
sign	O
of	O
the	O
elementary	O
product	O
needed	O
to	O
calculate	O
the	O
determinant	O
of	O
an	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
is	O
related	O
to	O
whether	O
the	O
corresponding	O
triangular	O
number	O
is	O
even	O
or	O
odd	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
number	O
of	O
inversions	O
in	O
the	O
permutation	B-Algorithm
for	O
the	O
only	O
nonzero	O
signed	O
elementary	O
product	O
of	O
any	O
n×n	O
anti-diagonal	B-Algorithm
matrix	I-Algorithm
is	O
always	O
equal	O
to	O
the	O
nth	O
such	O
number	O
.	O
</s>
