We know that any non-zero number has an inverse; for example has an inverse or . The inverse of the number is usually written or, more formally, by . This numerical inverse has the property that
We now show that an inverse of a matrix can, in certain circumstances, also be defined.
Given an square matrix then an square matrix is said to be the inverse matrix of if
where is, as usual, the identity matrix (or unit matrix) of the appropriate size.
Show that the inverse matrix of is
All we need do is to check that .
The reader should check that also.
We make three important remarks:
Non-square matrices do not have inverses.
The inverse of is usually written .
Not all square matrices have inverses.
Consider , and let be a possible inverse of .
Equate the elements of
to those of
and solve the resulting equations:
. Hence . This is not possible!
Hence, we have a contradiction. The matrix therefore has no inverse and is said to be a singular matrix . A matrix which has an inverse is said to be non-singular .
If a matrix has an inverse then that inverse is unique.
Suppose and are both inverses of . Then, by definition of the inverse,
Consider the two ways of forming the product
Hence and the inverse is unique .
There is no such operation as division in matrix algebra.
We do not write but rather
depending on the order required.
Assuming that the square matrix has an inverse then the solution of
the system of equations is found by pre-multiplying both sides by .