DEFINITION: A matrix is defined as an ordered rectangular array of
numbers. They can be
used to re present systems of linear equations , as will be explained below
Here are a couple of examples of different types of matrices:
Symmetric 
Diagonal 
Upper
Triangular 
Lower
Triangular 
Zero 
Identity 






And a fully expanded mxn matrix A, would look like this:
or
in a more compact form:
DEFINITION: Two matrices A and B can be added or subtracted if and
only if their
dimensions are the same (i.e. both matrices have the identical amount of rows
and columns.
Take:
Addition
If A and B above are matrices of the same type then the sum is found by adding
the
corresponding elements
Here is an example of adding A and B together
Subtraction
If A and B are matrices of the same type then the subtraction is found by
subtracting the
corresponding elements
Here is an example of subtracting matrices
Now, try adding and subtracting your own matrices
Matrix Multiplication
DEFINITION: When the number of columns of the first matrix is the same
as the number of
rows in the second matrix then matrix multiplication can be performed.
Here is an example of matrix multiplication for two 2x2 matrices
Here is an example of matrices multiplication for a 3x3 matrix
Now lets look at the nxn matrix case, Where A has dimensions mxn, B has
dimensions nxp.
Then the product of A and B is the matrix C, which has dimensions mxp. The ij^{th}
element of
matrix C is found by multiplying the entries of the i^{th}
row of A with the
corresponding entries
in the j^{th}
column of B and summing the n terms. The elements of C are:
Note: That AxB is not the same as BxA
Now, try multiplying your own matrices
Transpose of Matrices
DEFINITION: The transpose of a matrix is found by ex changing rows for columns
i.e.
Matrix A = (a_{ij}) and the transpose of A is:
A^{T}=(a_{ji}) where j is the column number and i is the row number of matrix A.
For example, The transpose of a matrix would be:
In the case of a square matrix (m=n), the transpose can be used to check if a
matrix is
symmetric. For a symmetric matrix A =
A^{T}
Now try an example
DEFINITION: Determinants play an important role in finding the inverse of
a matrix
and also in solving systems of linear equations. In the fol lowing we assume we
have a
square matrix (m=n). The determinant of a matrix A will be denoted by det(A) or
A.
Firstly the determinant of a 2x2 and 3x3 matrix will be introduced then the nxn
case will
be shown.
Determinant of a 2x2 matrix
Assuming A is an arbitrary 2x2 matrix A, where the elements are given by:
then the determinant of a this matrix is as follows:
Now try an example of finding the determinant of a 2x2 matrix yourself
Determinant of a 3x3 matrix
The determinant of a 3x3 matrix is a little more tricky and is found as follows
( for this case assume
A is an arbitrary 3x3 matrix A, where the elements are given below)
then the determinant of a this matrix is as follows:
Now try an example of finding the determinant of a 3x3 matrix yourself
Determinant of a nxn matrix
For the general case, where A is an nxn matrix the determinant is given by:
Where the coefficients
are given by the relation
where
is the determinant of the (n1) x (n1) matrix that is obtained by deleting row
i and
column j. This coefficient
is also called the cofactor of a_{ij}.
The Inverse of a Matrix
DEFINITION: Assuming we have a square matrix A, which is nonsingular (
i.e. det(A)
does not equal zero ), then there exists an nxn matrix A^{1} which is called the
inverse of A,
such that this property holds :
AA^{1}= A^{1}A = I where I is the identity matrix.
The inverse of a 2x2 matrix
Take for example a arbitury 2x2 Matrix A whose determinant (adbc) is not equal
to zero
where a,b,c,d are numbers, The inverse is:
Now try finding the inverse of your own 2x2 matrices:
The inverse of a nxn matrix
The inverse of a general nxn matrix A can be found by using the following
equation:
Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be
calculated by the following
method
•Given the nxn matrix A, define
to be the matrix whose coefficients are found by taking the determinant of the
(n1) x (n1)
matrix obtained by deleting the i^{th} row and j^{th} column of A. The terms of B
(i.e. B = b_{ij}) are
known as the cofactors of A.
•And define the matrix C, where
•The transpose of C (i.e C^{T}) is called the adjoint of matrix A.
Lastly to find the inverse of A divide the matrix C^{T} by the
determinant of A to give its inverse.
Now test this method with finding the inverse of your own 3x3 matrices
DEFINITION: A system of linear equations is a set of equations with n
equations and n
unknowns, is of the form of
The unknowns are denoted by x_{1},x_{2},...x_{n} and
the coefficients (a's and b's above) are
assumed to be given. In matrix form the system of equations above can be written
as:
A simplified way of writing above is like this ; Ax = b
Now, try putting your own equations into matrix form by clicking here:
After looking at this we will now look at two methods used to solve matrices
these are
•Inverse Matrix Method
•Cramer's Rule
Inverse Matrix Method
DEFINITION: The inverse matrix method uses the inverse of a matrix to
help solve a
system of equations, such like the above Ax = b. By
premultiplying both sides of this
equation by A^{1} gives:
or alternatively this gives
So by calculating the inverse of the matrix and multiplying this by the
vector b we can
find the solution to the system of equations directly. And from earlier we found
that the
inverse is given by
From the above it is clear that the existence of a solution depends on the
value of the
determinant of A. There are three cases:
1. If the det(A) does not equal zero then solutions exist using
2. If the det(A) is zero and b=0 then the solution will be not be unique or does
not
exist.
3. If the det(A) is zero and b=0 then the solution can be x = 0 but as in 2. is
not unique
or does not exist.
Looking at two equations we might have that
Written in matrix form would look like
and by rearranging we would get that the solution would look like
Now try solving your own two equations with two unknowns
Similarly for three simultaneous equations we would have:
Written in matrix form would look like
and by rearranging we would get that the solution would look like
Now try solving your own three equations with three unknowns
Cramer's Rule
DEFINITION: Cramer's rules uses a method of determinants to solve systems
of
equations. Starting with equation below,
The first term x_{1} above can be found by replacing the first column of A by
.
Doing this we obtain:
Similarly for the general case for solving x_{r} we replace the r^{th}
column of A by
and expand the determinant
This method of using determinants can be applied to solve
systems of linear equations.
We will illustrate this for solving two simultaneous equations in x and y and
three
equations with 3 unknowns x, y and z.
Two simultaneous equations in x and y
To solve use the following:
or simplified:
Now try solving two of your own equations
Three simultaneous equations in x, y and z
ax+by+cz = p
dx+ey+fz = q
gx+hy+iz = r
To solve use the following:
Now try solving your own three equations