Matrix Inverse
The inverse of an n x n matrix
is denoted by
. The inverse is defined so that
Where In is the identity matrix.
It should be stressed that only square matrices have inverses proper- however, a
matrix of any dimension may have ``left'' and ``right'' inverses (which will not be
discussed here).
A precondition for the existence of the matrix inverse
(i.e. the matrix is invertible) is
that det(A) does NOT equal zero. (the determinant is nonzero), the reason for which we
will see in a second.
The general form of the inverse of a matrix
is
where
is the adjoint of . This can be thought of as a generalization of the 2 x 2
formula given in the next section. However, due to the inclusion of the determinant in
the expression, it is impractical to actually use this to calculate inverses.
This general form also explains why the determinant must be nonzero for invertability;
as we are dividing through by its value.
Calculating the Inverse By Hand
An easy way to calculate the inverse of a matrix by hand is to form an augmented
matrix [ A | In ] from and In, then use Gaussian elimination to transform the left half
into . At the end of this procedure, the right half of the augmented matrix will be
(that is, you will be left with [ In |
]).
FORMULAS:
2-by-2 case:
For the 2 x 2 case, the general formula reduces to a memorable shortcut.
The inverse
where
is always
is simply ad - bc.
3-by-3 case:
Transpose of Matrices
DEFINITION: The transpose of a matrix is found by exchanging rows for columns
i.e. Matrix A = (aij) and the transpose of A is:
AT=(aji) 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 = AT
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 (n-1) x (n-1) matrix obtained by deleting the ith row and jth column of A.
The terms of B (i.e. B = bij) are known as the cofactors of A.
And define the matrix C, where

.
The transpose of C (i.e CT) is called the adjoint of matrix A.
Lastly to find the inverse of A divide the matrix CT by the determinant of A to give
its inverse.
Now test this method with finding the inverse of your own 3x3 matrices
Cramer's Rule
DEFINITION: Cramer's rules uses a method of determinants to solve
systems of equations. Starting with equation below,
The first term x1 above can be found by replacing the first column of A
by
. Doing this we obtain:
Similarly for the general case for solving xr we replace the rth 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:
and
or simplified:
and