TABLE OF CONTENTS This appendix covers the simple algebra of matrices, and some properties of eigenvalues and eigenvectors. The calculation of eigenvalues and eigenvectors is the main topic of Chapter 27.
For the most part, we will only need to consider the algebra of 2 2 matrices,
Addition of two matrices is component by component, so that
while multiplication is given by
We can write this more compactly by saying that
where [ 
] ij is the entry in the ith row and jth column of the matrix .
One special matrix is the identity matrix,
which has the property that 
= = for any 2 2 matrix .
A matrix is said to be invertible, or non-singular, if there is another matrix 
such that
The matrix
is invertible if and only if its determinant, det( ), given by
is not equal to zero, and then
In general we can calculate the product when is an n m matrix and is an m k matrix (the columns of have to match the rows of ). In particular this allows us to calculate x if is a 2 2 matrix and x = ( x 1, x 2) is a two component vector,
We can also write this more compactly as
where v i indicates the
