<s>
In	O
mathematics	O
,	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
whose	O
entries	O
are	O
all	O
integers	O
.	O
</s>
<s>
Examples	O
include	O
binary	B-Algorithm
matrices	I-Algorithm
,	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
,	O
the	O
matrix	B-Architecture
of	O
ones	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
and	O
the	O
adjacency	B-Algorithm
matrices	I-Algorithm
used	O
in	O
graph	O
theory	O
,	O
amongst	O
many	O
others	O
.	O
</s>
<s>
Integer	B-Algorithm
matrices	I-Algorithm
find	O
frequent	O
application	O
in	O
combinatorics	O
.	O
</s>
<s>
are	O
both	O
examples	O
of	O
integer	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Invertibility	O
of	O
integer	B-Algorithm
matrices	I-Algorithm
is	O
in	O
general	O
more	O
numerically	O
stable	O
than	O
that	O
of	O
non-integer	O
matrices	O
.	O
</s>
<s>
The	O
determinant	O
of	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
is	O
itself	O
an	O
integer	O
,	O
thus	O
the	O
numerically	O
smallest	O
possible	O
magnitude	O
of	O
the	O
determinant	O
of	O
an	O
invertible	O
integer	B-Algorithm
matrix	I-Algorithm
is	O
one	O
,	O
hence	O
where	O
inverses	O
exist	O
they	O
do	O
not	O
become	O
excessively	O
large	O
(	O
see	O
condition	B-Algorithm
number	I-Algorithm
)	O
.	O
</s>
<s>
Theorems	O
from	O
matrix	B-Architecture
theory	I-Architecture
that	O
infer	O
properties	O
from	O
determinants	O
thus	O
avoid	O
the	O
traps	O
induced	O
by	O
ill	B-Algorithm
conditioned	I-Algorithm
(	O
nearly	O
zero	O
determinant	O
)	O
real	O
or	O
floating	B-Algorithm
point	I-Algorithm
valued	O
matrices	O
.	O
</s>
<s>
The	O
inverse	O
of	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
is	O
again	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
if	O
and	O
only	O
if	O
the	O
determinant	O
of	O
equals	O
or	O
.	O
</s>
<s>
Integer	B-Algorithm
matrices	I-Algorithm
of	O
determinant	O
form	O
the	O
group	O
,	O
which	O
has	O
far-reaching	O
applications	O
in	O
arithmetic	O
and	O
geometry	O
.	O
</s>
<s>
The	O
intersection	O
of	O
the	O
integer	B-Algorithm
matrices	I-Algorithm
with	O
the	O
orthogonal	O
group	O
is	O
the	O
group	O
of	O
signed	O
permutation	O
matrices	O
.	O
</s>
<s>
The	O
characteristic	O
polynomial	O
of	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
has	O
integer	O
coefficients	O
.	O
</s>
<s>
Since	O
the	O
eigenvalues	O
of	O
a	O
matrix	B-Architecture
are	O
the	O
roots	O
of	O
this	O
polynomial	O
,	O
the	O
eigenvalues	O
of	O
an	O
integer	B-Algorithm
matrix	I-Algorithm
are	O
algebraic	O
integers	O
.	O
</s>
<s>
Integer	B-Algorithm
matrices	I-Algorithm
are	O
sometimes	O
called	O
integral	B-Algorithm
matrices	I-Algorithm
,	O
although	O
this	O
use	O
is	O
discouraged	O
.	O
</s>
