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The '2 x 2 real matrices' are the linear mappings of the Cartesian coordinate system into itself by the rule
:
(ax + by, cx + dy).
The set of all such real matrices is denoted by M(2,R). Two matrices ''p'' and ''q'' have a sum ''p'' + ''q'' given by matrix addition. The product matrix ''p q'' is formed from the dot product of the rows and columns of its factors through matrix multiplication. For
: let $quad\; q^\{$
★ } =egin{pmatrix}d & -c \ -b & a end{pmatrix}.
Then ''q q'' 
★ = (''ad'' − ''bc'') ''I'', where ''I'' is the 2 x 2 identity matrix. The real number ''ad'' − ''bc'' is called the determinant of ''q''. Evidently when ''ad'' − ''bc'' ≠ 0, ''q'' is an invertible matrix and ''q''⁻¹ = ''q'' 
★  / (''ad'' − ''bc''). The collection of all such invertible matrices constitutes the general linear group GL(2,R). In terms of abstract algebra, the set of 2 by 2 real matrices and their associated addition and multiplication operators forms a ring, and GL(2,R) is its group of units. M(2,R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphic to the coquaternions, but has a different profile.

Contents

Profile

Equi-areal mapping

Profile

Within M(2,R), the multiples by real numbers of the identity matrix ''I'' may be considered a real line. Since every matrix lies in a commutative subring of M(2,R) that includes this real line, the whole ring can be profiled by such subrings. Toward this end one needs matrices ''m'' such that ''m''² ∈ { −''I'', 0, ''I'' } to form planes
P_''m'' = {''x I'' + ''ym'' : ''x'', ''y'' ∈ R}, which are in fact commutative subrings.
The square of the generic matrix is
:
which is diagonal when ''a'' + ''d'' = 0. Thus we assume ''d'' = −''a'' when looking for ''m'' to form commutative subrings. When ''mm'' = −''I'', then ''bc'' = −1 − ''aa'', an equation describing an hyperbolic paraboloid in the space of parameters (''a'', ''b'', ''c''). In this case P_''m'' is isomorphic to the field of (ordinary) complex numbers. When ''mm'' = +''I'', ''bc'' = +1 − ''aa'', giving a similar surface, but now P_''m'' is isomorphic to the ring of split-complex numbers. The case ''mm'' = 0 arises when only one of ''b'' or ''c'' is non-zero, and the commutative subring P_''m'' is then a copy of the dual number plane.

Equi-areal mapping

First transform one differential vector into another:
:$$
egin{align}
(du, dv) & {} = (dx, dy) egin{pmatrix}p & r\ q & s end{pmatrix} \
& {} = (p, dx + q, dy , r, dx + s, dy).
end{align}

Areas are measured with ''density'' $dx\; wedge\; dy$ , a differential 2-form which involves the use of exterior algebra. The transformed density is
:$$
egin{align}
du wedge dv & {} = 0 + ps dx wedge dy + qr dy wedge dx + 0 \
& {} = (ps - qr) dx wedge dy = (det g) dx wedge dy.
end{align}
Thus the equi-areal mappings are identified with
SL(2,R) = {''g'' ∈ M(2,R) : det(''g'') = 1}, the 'special linear group'. Given the profile above, every such ''g'' lies in a commutative subring P_''m'' representing a type of complex plane according to the square of ''m''. Since ''g g'' 
★  = ''I'', one of the following three alternatives occurs:

★ ''mm'' = −''I'' and ''g'' is on a circle of Euclidean rotations; or

★ ''mm'' = ''I'' and ''g'' is on an hyperbola of squeeze mappings; or

★ ''mm'' = 0 and ''g'' is on a line of shears.