<s>
In	O
linear	B-Language
algebra	I-Language
,	O
an	O
augmented	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
obtained	O
by	O
appending	O
the	O
columns	O
of	O
two	O
given	O
matrices	O
,	O
usually	O
for	O
the	O
purpose	O
of	O
performing	O
the	O
same	O
elementary	O
row	O
operations	O
on	O
each	O
of	O
the	O
given	O
matrices	O
.	O
</s>
<s>
For	O
a	O
given	O
number	O
of	O
unknowns	O
,	O
the	O
number	O
of	O
solutions	O
to	O
a	O
system	O
of	O
linear	O
equations	O
depends	O
only	O
on	O
the	O
rank	O
of	O
the	O
matrix	B-Architecture
representing	O
the	O
system	O
and	O
the	O
rank	O
of	O
the	O
corresponding	O
augmented	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Specifically	O
,	O
according	O
to	O
the	O
Rouché	O
–	O
Capelli	O
theorem	O
,	O
any	O
system	O
of	O
linear	O
equations	O
is	O
inconsistent	O
(	O
has	O
no	O
solutions	O
)	O
if	O
the	O
rank	O
of	O
the	O
augmented	B-Algorithm
matrix	I-Algorithm
is	O
greater	O
than	O
the	O
rank	O
of	O
the	O
coefficient	O
matrix	B-Architecture
;	O
if	O
,	O
on	O
the	O
other	O
hand	O
,	O
the	O
ranks	O
of	O
these	O
two	O
matrices	O
are	O
equal	O
,	O
the	O
system	O
must	O
have	O
at	O
least	O
one	O
solution	O
.	O
</s>
<s>
An	O
augmented	B-Algorithm
matrix	I-Algorithm
may	O
also	O
be	O
used	O
to	O
find	O
the	O
inverse	O
of	O
a	O
matrix	B-Architecture
by	O
combining	O
it	O
with	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
To	O
find	O
the	O
inverse	O
of	O
C	O
we	O
create	O
(	O
C|I	O
)	O
where	O
I	O
is	O
the	O
2×2	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
We	O
then	O
reduce	O
the	O
part	O
of	O
(	O
C|I	O
)	O
corresponding	O
to	O
C	O
to	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
using	O
only	O
elementary	O
row	O
operations	O
on	O
(	O
C|I	O
)	O
.	O
</s>
<s>
the	O
right	O
part	O
of	O
which	O
is	O
the	O
inverse	O
of	O
the	O
original	O
matrix	B-Architecture
.	O
</s>
<s>
In	O
this	O
example	O
the	O
coefficient	O
matrix	B-Architecture
has	O
rank	O
2	O
while	O
the	O
augmented	B-Algorithm
matrix	I-Algorithm
has	O
rank	O
3	O
;	O
so	O
this	O
system	O
of	O
equations	O
has	O
no	O
solution	O
.	O
</s>
<s>
As	O
used	O
in	O
linear	B-Language
algebra	I-Language
,	O
an	O
augmented	B-Algorithm
matrix	I-Algorithm
is	O
used	O
to	O
represent	O
the	O
coefficients	O
and	O
the	O
solution	O
vector	O
of	O
each	O
equation	O
set	O
.	O
</s>
<s>
Note	O
that	O
the	O
rank	O
of	O
the	O
coefficient	O
matrix	B-Architecture
,	O
which	O
is	O
3	O
,	O
equals	O
the	O
rank	O
of	O
the	O
augmented	B-Algorithm
matrix	I-Algorithm
,	O
so	O
at	O
least	O
one	O
solution	O
exists	O
;	O
and	O
since	O
this	O
rank	O
equals	O
the	O
number	O
of	O
unknowns	O
,	O
there	O
is	O
exactly	O
one	O
solution	O
.	O
</s>
