Understanding Engineering Mathematics

13.1 An overview of matrices and determinants

Everywhere in engineering and science we are constantly having to deal with situations
where there are many variables. Matrices provide the mathematical shorthand for dealing
with a number of quantities at the same time – basically a generalisation of vectors (indeed
the language is similar in the two topics). Matrices are used in modelling complex mechan-
ical systems, control systems, electronic circuits, optical systems, economic and financial
systems, network problems, robotics, and much more besides.
Amatrixis essentially a rectangular array of numbers, each independent from each
other – only their position in the array is significant. In matrix algebra the array itself
is treated as a single mathematical object – just like a vector. Rules have been invented
for combining such arrays, much like the rules of arithmetic, but with some significant
differences. Thus, matrices can be added and subtracted. They can also be multiplied,
but the commutative rulexy=yxno longer holds in general – ifA, Bare two matrices
thenABandBAare not necessarily the same thing. Matrices can also be ‘divided’,
but care is needed, as always with division. The rules of multiplication and division of
matrices are constructed so that we can use matrices in such things as solving systems of
equations.
Remember that in complex numbers we found it useful to introduce a real number – the
modulus,|z|– associated with a complex number,z, in order to define division byz.We
do a similar, but more general thing with matrices. Thus, adeterminantis a number
associated with a given matrix or array. It is evaluated in a certain way from the elements
of the matrix and gives us useful information about it. Determinants enable us to define
a series of numbers associated with a (square) matrix, called itseigenvalues.Theseare
extremely important both mathematically and for applications. If for example the matrix
describes a particular control system then the eigenvalues of that matrix tell us important
things about the stability of the control system.

13.2 Definition of a matrix and its elements

Consider the system of equations

2 x+y= 3

( 13. 1 )

x− 2 y=− 1

In a sense the important information here is the set of coefficients, which we can arrange
in two arrays:

A=

[

21

1 − 2

]

b=

[

3

− 1

]

Such arrays are calledmatrices.Ais referred to as a 2 ×2matrix, because it has two
rows and two columns. Similarlybis a 2×1 matrix, orcolumn vector. The numbers
in the array are called itselements. Their positions are important, and note that signs