Matrices

A matrix (plural ‘matrices’) is defined as a rectangular array of numbers. A matrix
is usually represented by a set of numbers set in square brackets [ ] as in the
example below.

From the above example, you should observe that the entries in the matrix are arranged
in rows and columns

The entries in a matrix can be any number (real or imaginary) or even variables.
In the example above, a,b,c,d can be numbers or variables. The name of the
matrix is always capitalized; in the above example, the matrix is named A.

A matrix is defined by the number of rows and columns that it contains and this
is known as the size of the matrix.

The size of a matrix is denoted by n x m (pronounced as n by m)
where n represents the number of rows in the matrix and m represents
the number of rows. The smallest value that n and m can have is 1.
You can’t have a matrix with zero rows or zero columns. The size of the matrix can
range from 1 x 1 to ∞ x ∞.

When a matrix has only one row, it is referred to as a row vector while a
matrix with only column is referred to as a column vector. Consequently,
any row or column in a matrix can be referred to as a row vector or a column vector.

The standard notation used to refer to rows and columns of a matrix is the i,j
notation; where i represents a specific row of a matrix and j
represents a specific column, always row first then column. This notation allows
one to refer to a specific entry in the matrix by defining its exact position in
the form

For example

In the example above, we already defined the name of the matrix as A, and
we can see that the matrix is of size 2 x 2 meaning that it has 2 rows and 2 columns.
These rows and columns are numbered as row 1 (first row consisting of entries a
and b) which is the top most row; and row 2 (second row consisting of entries c
and d) which follows directly below. If there had been a more rows, they would have
been numbered subsequently in a increasing order. The columns of the matrix follow
a similar pattern; the left most column is referred to as column 1 (first column
consisting of entries a and c); and the next column is column 2 (second column consisting
of entries b and d), with subsequent columns numbered in a similar increasing order.

Each of the entries in A can be defined as follows

which means that a is located in the first row and first column of matrix
A, and so to find a, you look for where the first row intersects the
first column.

meaning that b is located in the position where the first row intersects
the second column.

meaning that c is located in the position where the second row intersects
the first column.

meaning that d is located in the position where the second row intersects
the second column.

It is important to remember that the notation refers to row first then column.

Matrix Algebra

Matrices can be equated, added, subtracted and multiplied but there rules as to
how and when this can be done. Matrices CAN’T be divided ordinarily as you would
divide numbers. Dividing matrices involved finding the inverse of the matrix
but more on this later. The next page deals with matrix equality
which is the first aspect of matrix algebra that we’ll deal with.