Introduction
to Matrices (plural for Matrix)
(This
material is experted from http://www.tutor.ms.unimelb.edu.au/matrix/matrix_basics.html
etc)
DEFINITION
A matrix is a rectangular array of items. In our context the items
will typically be numbers.
EXAMPLES. A and B below are examples of matrices.
A = 

B = 

Most books write the array inside large brackets, but that is
difficult here, so we will use a border right around instead.
TERMINOLOGY AND NOTATION
§ If a
matrix has m rows and n columns we say it is an mxn
matrix or that the SIZE of the matrix is mxn.
In the example above, A is a 3x2 matrix.
§ When m =
n we say the matrix is SQUARE. In the example above, B is a square matrix and has size 2x2.
§ The
numbers in the array are called ELEMENTS or ENTRIES of the
matrix.
§ If we
want to write a general matrix A, we denote the element in row i and column j as a_{ij}.
So a shorthand way of writing A is A = (a_{ij})
where i = 1, 2, ...m and j = 1, 2, ...n and by this
we mean
A = 

§ For A in
the example above, a_{22} = 0, a_{31} = 2. For B in the
example above, b_{12} = 9, b_{22} = 8.
§ If a
matrix has only one row it is called a ROW MATRIX or ROW VECTOR; If a
matrix has only one column it is called a COLUMN MATRIX or COLUMN
VECTOR.
EXAMPLES. C below is an example of a
row matrix; D below is a column matrix.
C = 

D = 

BASIC PROPERTIES AND
OPERATIONS
EQUALITY
Two matrices are equalif and only if
they have the same size and corresponding elements are equal.
EXAMPLE. When does

= 

? 
The two
matrices have the same size. To be equal we also need
a = 1, b = 2, c = 3, d = 4. 
ZERO MATRIX
A zero matrix is a matrix with all its entries zero. We denote the
zero matrix by 0 if there is no confusion about size, otherwise we use 0_{mxn}.
EXAMPLES.
0_{3x2} = 

0_{1x2} = 

OPERATIONS
ADDITION (SUBTRACTION)
If A and B are the same size we can add
(subtract) them and we do so by adding (subtracting) corresponding elements.
EXAMPLES.
(i)

+ 

= 

= 

(ii)

+ 

is not defined because the matrices are different sizes.
PROPERTIES OF THE ADDITION
OPERATION
If A, B, C are all mxn matrices (i.e.
all the same size) then
A + B = B
+ A
A + (B + C) = (A + B) + C
A + 0_{mxn} = A.
SCALAR MULTIPLICATION
If A is any matrix and p is any real number, the product pA is the matrix obtained by multiplying each element of A
by p.
EXAMPLE.
3 

= 

PROPERTIES OF THE SCALAR
MULTIPLICATION OPERATION
If p and q are real numbers and A and B are matrices of the same
size, then
p(A + B) = pA + pB
(p + q)A = pA + qA
pqA = p(qA)
TRANSPOSE OF A MATRIX
If A is an mxn matrix then the transpose
of A, denoted A^{T} or A^{t} or
sometimes A^{'}, is the nxm matrix obtained
by interchanging the rows and columns of A.
Thus the first row becomes the first column, the second row
becomes the second column, etc.
EXAMPLE. Find the transpose of

The transpose is

PROPERTIES OF THE TRANSPOSE
OPERATION
If A and B are matrices of the same size, then
(A^{T})^{T} = A
(A + B)^{T} = A^{T} + B^{T}.
MATRIX MULTIPLICATION
How do we multiply a matrix by a matrix? The definition is not the
natural thing one would first think of, but, strange as it at first seems, it
is what has been found to be very useful.
DEFINITION
Given two matrices A = (a_{ij})
and B = (b_{ij}) we can only find AB if the
number of columns of A is the same as the number of rows of B.
Suppose that A is mxn and B is nxp, (i.e. the number of columns of A is the same as the
number of rows of B), then C = AB is an mxp matrix
where the elements of C, c_{ij}, are given by
c_{ij} = a_{i1}b_{1j}
+ a_{i2}b_{2j} + ...+ a_{in}b_{nj}
i.e. To get the ij element of C, multiply
the elements of row i from A with the corresponding
elements of column j from B and add.
EXAMPLE. Find AB and BA if possible, where
A = 

B = 

A is 2x2 and B is 2x2, so AB exists and BA exists.
AB = 

= 

(We have used the definition as follows:For example for the 1,2 entry of AB: Take row
1 of A and column 2 of B. Multiply corresponding elements and add: 1x6 + 2x8.)
BA = 

= 

EXAMPLE. Find AB and BA if possible, where
A = 

B = 

A is 3x2 and B is 2x4, so AB exists and is of size 3x4. Since B is
2x4 and A is 3x2, 4 does not equal 3, so BA does not exist.
AB = 

= 

PROPERTIES OF MATRIX
MULTIPLICATION
§ VERY
IMPORTANT: As the previous examples show AB and BA are not usually equal,
and in fact one may exist whilst the other does not. Thus it is VERY important
to be precise in the order in which we write the matrices that we are
multiplying. Take particular notice of this in the properties below.
§ If the
sizes of the matrices below are such that the following matrix multiplications
are defined, then
(AB)C = A(BC)
(A + B)C = AC + BC
C(A + B) = CA + CB
0A = A0 = 0.
POWERS OF SQUARE MATRICES
We define
A^{2}
= AA, A^{3} = AAA, etc. Note that these are only defined if A is
square.
MATRIX IDENTITIES: EVEN
MATRICES NEED AN IDENTITY!
A SQUARE matrix with all 1's on the main diagonal (from the upper
left corner to the lower right corner) and 0's elsewhere is called an identity
matrix. If it is an nxn matrix, we denote it I_{n}, but if the size is clear from the context we
often don't bother to write the n. E.g.
I_{2} = 

I_{3} = 

I plays a similar role in matrix theory
to the role 1 plays in real number multiplication. For matrices we have, if A
is an nxn matrix, then
A I_{n} = I_{n}A = A.