Consider next the system of two equations in two unknowns
a 11 xþa 12 y¼c 1
a 21 xþa 22 y¼c 2
The subscripts of the unknowns coefficientsaindicate row 1, column 1, row 1,
column 2, etc. We’ll see that using matrices the solutions (the values ofxandy) can
be expressed in a way analogous to that for the equationax¼b.
A matrix is a rectangular array of “elements” (numbers, derivatives, etc.) that
obeys certain rules of addition, subtraction, and multiplication. Curved or angular
brackets are used to denote a matrix:
12
72
^5
2
0
0
B
@
1
C
A ðÞ^00742 & 2 matrix 3 & 1 matrix 1 & 4 matrixor:
12
72
5
2
0
2
4
3
(^5) ½ 0074
Do not confuse matrices with determinants (below), which are written with
straight lines, e.g.
12
73
is a determinant, not a matrix. This determinant represents the number 1& 3 " 2 &
7 ¼ 3 " 14 ¼"11. In contrast to a determinant, a matrix is not a number, but rather
an operator (although some would consider matrices to be generalizations of
numbers, with e.g. the 1&1 matrix (3)¼3). An operator acts on a function (or a
vector) to give a new function, e.g.d/dxacts on (differentiates)f(x) to givef^0 (x):
d
dx
fðxÞ¼
dfðxÞ
dx
¼f^0 ðxÞ
and the square root operator acts ony^2 to givey. When we have done matrix
multiplication you will see that a matrix can act on a vector and rotate it through an
angle to give a new vector.
Let’s look at matrix addition, subtraction, multiplication by scalars, and matrix
multiplication (multiplication of a matrix by a matrix).
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 109