In linear algebra, matrix inverse holds a special place because there is not division in matrix algebra. You cannot divide two matrices. Fortunately, the division is possible when a matrix is multiplied with its inverse which is unique.
The inverse is not possible with just any kind of matrix, a matrix must be square and invertible and the reasons are explained in this article along with several identities and examples involving inverse matrices.
An inverse matrix can also be used for finding the solution for system of linear equations, that is, where
is augmented matrix,
is the solution vector and
is the constant vector.
What Is The Need For Inverse?
Inverse means opposite of some operation performed and the result obtained is identity of that operation.
For example,
Additive identity
------------------
If
then
is additive identity.
but
(additive identity)
Therefore, subtraction is inverse operation of addition.
Multiplicative identity
-----------------------
If
then
is multiplicative identity because it gives
as result.
but
( multiplicative identity)
Therefore, multiplying with reciprocal or division is inverse operation of multiplication.
The same idea can be extended to matrix since we are unable to divide two matrices directly. If is a square matrix and invertible , then find an inverse matrix
such that multiplying it with
will give an identity matrix
of same order.

For example,
Let
be a square matrix of order 2 x 2. The inverse of matrix
is
.
Compute the determinant
Add negative to following elements in matrix
Swap the positive entries
Multiply the above result with
Let us now verify whether

Why Square Matrix ?
The inverse deals with negative power such as , a non-square matrix is cannot be used because it is undefined( cannot multiply).
The second reason for using square matrix is the identity matrix. An identity matrix is a square matrix only. A product of non-square matrix with its inverse will not result in an identity matrix.
If a square matrix has inverse matrix
such that

Then the matrix is called invertible matrix and matrix
is its inverse. If there is no
for matrix
, then it is called Singular matrix.
A matrix is singular and has no inverse if its determinant is 0. You will learn about determinants in future lessons.
Suppose
is a singular matrix of order 2 x 2.
In the same manner, determinants of higher order matrices is found.
Therefore, only square matrix is used to find inverse which is also a square matrix of size .
Uniqueness Of Inverse Matrix
If a square matrix is invertible, then it has exactly one inverse.
Proof :
Suppose that there are two inverse and
for matrix
. We get
- (1)
- (2)
We know that any matrix multiplied by Identity matrix will result itself. Therefore, the following is true.
\\by (2)
\\ by associativity property
\\ by (1)
Therefore, inverse is unique.
Use Of Inverse Matrix
The purpose of using matrix is to solve for where
represents the augmented matrix obtained from the system of linear equations,
is the vector of unknowns or solution vector and
is the constant vector.
Multiply both sides by
. Note that the order of operation is important.
By
, we get

Let us try to solve a system of equation using above result where matrix is invertible and square. Suppose the system of equations have following equation.

Let
be a square and invertible augmented matrix of order 2 x 2 derived from the system of equations above. Therefore,
is as follows.

Let us find the inverse of matrix . But, first we must find the determinant of matrix
.

Change the sign and swap the positive entries. Then multiply it with to get the inverse of matrix
.
Swap the positive entries.
Multiply with

We need to verify if this is correct inverse .

The inverse is correct and compute the value of solution vector using
in the same order.
where
and

We must verify the solution by substitution in the system of linear equations.

Similarly, we can verify some other interesting results in the following section.
Other Interesting Results
In this section, we will verify some other interesting results concerning inverse matrices.
(a) Product of two or more invertible matrices are invertible matrix.
//order is important
Proof:
Let and
be two invertible matrices of order
. Then
. If matrix
is invertible, then its inverse is
.
Therefore,
//because

Example #1
Let
and
be
invertible matrix.
Let
then

We will now find the inverse of the product matrix , that is,
. First compute the determinant of
matrix.

Now change value of element and
to negative in matrix
. Then swap the remaining positive values. Multiply the resultant matrix with
.

We must find the product matrix .

Therefore,
(b) Inverse of inverse matrix is the original matrix.
Let
be a
invertible matrix. Let
. Therefore, inverse of matrix
is the matrix
where
.
We know that
.
Therefore,

Example #2
Let
be a
invertible matrix.
The inverse of the matrix
is
Let us take inverse of inverse matrix
Change signs and swap positive values in
Multily above result with
.
Therefore, .
(c) If non-negative power of a invertible square matrix is
, then negative power of invertible square matrix is

Example #3
Let
be a invertible square matrix of order
. Let
be a positive integer.
Let
be the inverse matrix for
.
But we know that

Therefore, .
(d) If is a non-zero scalar and
is invertible square matrix, then

Proof:
We know that and also following algebraic identities applies in the case of matrix multiplication with scalars.
(1)
(2)
where
and
are defined matrices.
Using equation
we get
Using equation (2)
Using equation (1)
Therefore, is true.
Example #4
Let
and matrix
is invertible and order 2 x 2.
Multiply
with matrix
and take inverse.
Take determinant of the matrix
Take negative of
and
and swap positive values. Multiply with
(3)
We must compute the value of
(4)
(5)
(e) If is an invertible matrix of order
then the transpose
is also invertible and equal to transpose of inverse matrix
.

Example #5
Let
matrix of order
.
Inverse of Transpose
Transpose of Inverse

Therefore, .
In this article, we explained why and what are inverse of matrix. Next, we discuss how to obtain inverse of small to large invertible matrices using different available methods.