Evaluation of Inverse of a Square Matrix Using Elementary Transformations

Do you find it difficult to evaluate the inverse of a square matrix? Well, you will find it easier if you take help of elementary transformations. In other words, this method is also called the Gauss – Jordan method. It is not tough to understand elementary transformations. All you have to do is, go through this article thoroughly and you will find all your doubts solved here. 

You will get to know about some useful terms like elementary operations, elementary matrix, identity matrix, etc .Let us start our discussion with a brief definition of square matrix. When you study matrices, you will find that a matrix contains some rows and some columns as well. Any matrix having the same number of rows and columns is called a square matrix. To get the inverse of a square matrix, several techniques can be followed. However, the easiest and fruitful method of evaluating the inverse of a square matrix is elementary transformation. Come on let us have a quick look upon the techniques to be followed in terms of evaluating inverse of a square matrix through elementary transformations.

   Let us have an example,

      A = ( 4       8       0

               7       6       2

               1       3       9)

   Here, A is a square matrix of order 3. That means the matrix A has three rows and three columns.

What Is The Elementary Transformation Of A Matrix?

This particular procedure refers to some minor transformations in rows and columns of a matrix. Describing the variants of row operation can easily make the steps of elementary transformation clear. Column transformation operations can be done easily with those simple steps. The combined procedure of transforming rows and columns of a matrix is called elementary transformation.

Well, the process of elementary transformation consists of three different operations. The operations of elementary transformation are contained in the below list.

1. Rearranging two rows (can be performed the same with the columns too).
2. Multiplying all elements of a particular row with a certain constant (similar operation can be done with the column elements).
3. Multiplying all elements of a row with a certain constant and adding it to another row whose elements are also multiplied with a constant (in each case the constant is not equals to zero).

These are the three operations of elementary transformation of matrices.

How To Proceed With Elementary Transformation To Find The Inverse Of A Square Matrix?

We have already gathered a basic idea about elementary transformation of matrices. Now, it is time to understand the significance of elementary transformation in order to find the inverse of a square matrix.

To find the inverse of a square matrix with the help of elementary transformation we will need the following formula.

Working formula:-

Let us consider A is a non null square matrix of order “n”. We assume t8hat Ā denotes the inverse of matrix A. Then, the required formula to evaluate the inverse matrix of A is given by,

A.Ā = I(n)

[where I(n) is the identity matrix of order “n”]

With the help of the above formula we can easily derive the inverse matrix of any square matrix by following simple operations of elementary transformation. To derive an inverse matrix of any square matrix we have to follow these simple steps. In the following catalogue we are assuming the square matrix A, whose order is “n”. [We are assuming that Ā (which is unknown), is the inverse matrix of A]

1. Write down the matrix A on the left hand side of the equation and simply multiply Ā with it. Write down the identity matrix of the same order in the right hand side of the equation.
2. Apply elementary transformation operations as required. Keep the Ā unchanged as it is unknown.
3. The main objective of applying transformation operations is to transform the matrix A (in L.H.S.) into the identity matrix of order “n” while keeping Ā untouched.
4. At the end of elementary transformation, we will find a new matrix in the right hand side of the equation.
5. After all the operations are done, we can conclude that, Ā.I(n) i.e. Ā = the newly obtained matrix in R.H.S.

Thus, finding the inverse of any square matrix becomes easier with the help of elementary transformation methods.