Inverse of a Square Matrix

For any number a
(1)a = a(1) = 1
The number 1 is called the identity for multiplication

For an arbitrary matrix M, there is no identity matrix I such that
IM = MI = M
However, for square matrices -
     matrices where the number of rows is equal to the number of columns–
there is an identity matrix


Definition of Identity Matrix
The identity matrix for multiplication for the set of all square matrices of order n
is the square matrix of order n , denoted by I , with 1’s along the principal diagonal
(from upper left corner to lower right corner) and 0’s elsewhere

For example,

        is the identity matrix for all square matrices of order 3


     Identity Matrix Multiplication – see Example 1, page 747, of the textbook

In general, if M is a square matrix of order n and I is the identity matrix of order n , then

     IM = MI = M


Inverse of a Square Matrix
For any nonzero real number, there is a real number a^-1
called the multiplicative inverse of a
such that
     a^-1 a = 1


Definition of the the Inverse of a Square Matrix
If M is a square matrix of order n and if there exists a matrix M^-1
such that
     M^-1 M = M M^-1 = I

then M^-1 is called the multiplicative inverse of M or, simply, the inverse of M


Note:
The multiplicative inverse, a^-1 ,of a nonzero real number cane also be
written as 1/a
This notation is not used for matrix inverses



For example,
let us find the inverse, it it exists, for

     

By the definition of the multiplicative inverse, we must find the matrix
     

such that
     


So we write

            M            M^-1                  I
     


Working out the result of the matrix product on the left side of the equation,


     

By the definition of what it means for two matrices to be equal, we can write

                and     


Solving these two systems of linear equations, we get

     

So

     



We can check that

             M                  M^-1                  I                         M^-1            M
     


Note:
It seems that we must work out both M M^-1 and M^-1M to check our work,
but it is not necessary
It can be proved that if one of the equations

     M M^-1 = I or M^-1M = I

is satisfied, then the other equation is also satisfied
In other words, we need to versify only one of the equations


CAUTION:
Unlike nonzero real numbers, inverses do not always exist for nonzero square matrices
For example, an inverse does not exists for the matrix

     

The systems of linear equations

           and     
that we would need to solve are both inconsistent and have no solutions
In other words, an inverse does not exists

This method for finding the inverse, if it exists, can bet very tedious for matrices of order larger than 2.
However, we can use augmented matrices to make the process more efficient.


Inverse of a Square Matrix
If the augmented matrix
     
is transformed by row operations into the augmented matrix
     
then the resulting matrix
           is     
But if all 0’s appear in one or more rows to the left of the vertical line,
then does not exists

A matrix that does not have an inverse is called a singular matrix


Using the Augmented Matrix to find an Inverse Matrix –
     see Examples 2 - 4, pages 750 – 753, of the textbook


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