13 Matrices and Determinants
Matrices are rectangular arrays of numbers treated as mathematical entities in themselves
and satisfying an algebra that suits their particular application. In this respect they are
similar to vectors (Chapter 11), but whereas vectors are designed to reflect the behaviour
of physical quantities that have a direction associated with them, matrices are designed
to enable us to handle systems of linear equations of the sort introduced in Section 2.2.4.
Matrices have simple rules of addition and multiplication, but are complicated to invert, for
which it is convenient to introduce certain numbers, called determinants, that are associated
with matrices. Determinants also help us to define the eigenvalue problem, which is one
of immense importance in engineering and science.
Prerequisites
It will be helpful if you know something about:- solving simultaneous linear equations (48
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) - the principles and practice of elementary algebra (Chapter 12
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) - the sigma notation (102
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)
Objectives
In this chapter you will find:- definitions of matrices and determinants
- addition and subtraction of matrices
- multiplication of matrices
- zero and unit matrices
- properties of determinants
- the adjoint and inverse of a matrix
- Cramer’s rule for solving systems of linear equations
- eigenvalues and eigenvectors
Motivation
You may need the material of this chapter for:- solving systems of linear equations
- frequency analysis of coupled engineering systems
- solving distribution and scheduling problems in operational research
- solving systems of differential equations