Chemistry Reference
In-Depth Information
But we can also use the matrix product to generate a matrix for the
C
3
2
operation and then
apply this as a single operation, giving the same result:
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
001
100
010
001
100
010
b
1
b
2
b
3
010
001
100
b
1
b
2
b
3
b
2
b
3
b
1
⎝
⎠
⎝
⎠
⎝
⎠
=
⎝
⎠
⎝
⎠
=
⎝
⎠
(A5.12)
So, as we found for the case of the simpler basis in H
2
O, the matrix representation of
symmetry operations allows products of operations to be considered algebraically. This
matrix product approach can be extended to matrices of any size and so for any size of
basis.
A5.2 Matrices for Solving Sets of Linear Equations
Matrices also appear in the solution of problems in linear algebra because they provide a
compact way of discussing sets of equations. For example if we have three unknowns,
x
1
,
x
2
and
x
3
which conform to the similtaneous equations:
a
11
x
1
+
a
12
x
2
+
a
13
x
3
=
d
1
a
21
x
1
+
a
22
x
2
+
a
23
x
3
=
d
2
(A5.13)
a
31
x
1
+
a
32
x
2
+
a
33
x
3
=
d
3
where
a
ij
and
d
i
are constants; the equivalent matrix equation is
⎛
⎞
⎛
⎞
⎛
⎞
a
11
a
12
a
13
x
1
x
2
x
3
d
1
d
2
d
3
⎝
⎠
⎝
⎠
=
⎝
⎠
a
21
a
22
a
23
(A5.14)
a
31
a
32
a
33
for which we can use the shorthand
Ax
=
d
(A5.15)
This allows any algebraic manipulation to be carried out in the shorthand notation. For
example, if we wish to find the values of
x
i
in Equation (A5.13), then we can see
from Equation (A5.15) that, if we can find the inverse matrix of
A
, the solution will be
straightforward because
A
−
1
Ax
=
A
−
1
d
gives
x
=
A
−
1
d
(A5.16)
because
A
−
1
A
E
, the identity matrix.
To find the inverse matrix requires the introduction of the determinant. The determi-
nant is related to the square matrices we use as representations for symmetry operations,
but is a simple number formed in a systematic way from the elements of the matrix.
To distinguish the determinant from a matrix it is written enclosed in straight lines, rather
=
