<s>
A	O
Frobenius	B-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
square	B-Algorithm
matrix	I-Algorithm
from	O
numerical	B-General_Concept
mathematics	I-General_Concept
.	O
</s>
<s>
A	O
matrix	O
is	O
a	O
Frobenius	B-Algorithm
matrix	I-Algorithm
if	O
it	O
has	O
the	O
following	O
three	O
properties	O
:	O
</s>
<s>
The	O
inverse	O
of	O
a	O
Frobenius	B-Algorithm
matrix	I-Algorithm
is	O
again	O
a	O
Frobenius	B-Algorithm
matrix	I-Algorithm
,	O
equal	O
to	O
the	O
original	O
matrix	O
with	O
changed	O
signs	O
outside	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
.	O
</s>
<s>
The	O
term	O
Frobenius	B-Algorithm
matrix	I-Algorithm
may	O
also	O
be	O
used	O
for	O
an	O
alternative	O
matrix	O
form	O
that	O
differs	O
from	O
an	O
Identity	B-Algorithm
matrix	I-Algorithm
only	O
in	O
the	O
elements	O
of	O
a	O
single	O
row	O
preceding	O
the	O
diagonal	O
entry	O
of	O
that	O
row	O
(	O
as	O
opposed	O
to	O
the	O
above	O
definition	O
which	O
has	O
the	O
matrix	O
differing	O
from	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
in	O
a	O
single	O
column	O
below	O
the	O
diagonal	O
)	O
.	O
</s>
<s>
The	O
following	O
matrix	O
is	O
an	O
example	O
of	O
this	O
alternative	O
form	O
showing	O
a	O
4-by-4	O
matrix	O
with	O
its	O
3rd	O
row	O
differing	O
from	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
alternative	O
name	O
for	O
this	O
latter	O
form	O
of	O
Frobenius	O
matrices	O
is	O
Gauss	B-Algorithm
transformation	I-Algorithm
matrix	O
,	O
after	O
Carl	O
Friedrich	O
Gauss	O
.	O
</s>
<s>
They	O
are	O
used	O
in	O
the	O
process	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
to	O
represent	O
the	O
Gaussian	O
transformations	O
.	O
</s>
<s>
This	O
corresponds	O
to	O
one	O
of	O
the	O
elementary	O
operations	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
(	O
besides	O
the	O
operation	O
of	O
transposing	O
the	O
rows	O
and	O
multiplying	O
a	O
row	O
with	O
a	O
scalar	O
multiple	O
)	O
.	O
</s>
