Evaluation of inverse of a square matrix using elementary transformations

Introduction

In this article, we will introduce the evaluation of the inverse of a square matrix using elementary transformations. This topic educates learners about the inverse of a matrix by elementary operations, square matrices, and elementary transformations. 

 The inverse of a matrix

A matrix is a rectangular structure/array of symbols, characters, and numbers representing a set of information . The matrix elements are placed in columns and rows. The order of any matrix is  A x B, where A is the number of rows and B is the number of columns in that matrix. We can perform basic arithmetic operations of math on matrices like subtraction, multiplication and addition. A set of matrices are known to be equal if matrices have a similar order and their components are also the same. There is a distinction between the phrases ‘equal’ and ‘equivalent’. Two equivalent matrices are indicated by the use of the symbol ‘~’. A set of two matrices is known as equivalent if one of the matrices can be altered through the elementary transformation to get the second matrix. Elementary operation is a distinct type of operation that is executed on rows and columns of the matrices.

 Elementary operations

There are six elementary operations that are performed on a matrix, which include the three operations due to columns and three due to rows. These operations are called elementary operations. Elementary operations are performed on square matrices only. For example, 2×2 and 3×3 matrices.

Square matrix

A matrix that has the same number of columns and rows is called a square matrix. In mathematics, m × m matrix refers to the square matrix with the order m. The order of the resultant matrix when two square matrices are added or subtracted remains the same. Matrix multiplication of two square matrices A and B is only possible if they have the same order.

An example of a square matrix is:

X=    3   6     9

        4   -4     8  

        7       5    9

as number of rows=number of columns =3

 Elementary transformation

The procedures that are executed on columns and rows of the matrix to modify it into a varied figure so that the numbering becomes simpler is named as elementary transformation. ‘What are Elementary transformations’ theory is found in the Gaussian method of deducing the echelon figure of a matrix, linear equations, and other operations, including the matrix presentation of a strategy of equations. It is used for finding the determinants of the matrices, solving a system of linear equations, and inverse of the matrices as well. Elementary transformations are performed between any set of two matrices, with the condition that the order of the set is the same.

Elementary row transformation

Row transformations are done based on some sets of rules only. A person cannot execute any different kind of row operations other than the rules as follows. Three types of elementary transformations for rows are seen. 

3.   The rows of the matrix interchanging: In this system, the whole row of a matrix is replaced with a different row. It is depicted as Ri ↔ Rj, where j and i are two distinct row numbers.
4.   Gauging the entire row with a number that is non-zero: The whole row is multiplied with the number as before. It is depicted as Ri → k Ri, which implies that every component in the row is multiplied by the ‘k’ factor.
5.   The addition of one row to a different one scaled by a number that is non-zero: Each component of a row is replaced by a digit obtained by adding it to the scaled component of another row. It is depicted as Ri → Ri + k Rj.

Two matrices are known as row equivalents when one of the matrices can be found from the other matrix by the use of these three elementary row transformations.

Elementary column transformation

There are some sets of rules that should be followed while conducting transformations for columns. There are three types of aspects of elementary transformations for columns. There are no other authorized transformations other than these column transformations.

3. Switching the columns in the matrix: In this system, the whole column in a matrix is switched with a different column. It is depicted as  Ci ↔ Cj, where j and i are two varying column numbers.
4. Multiplying the entire column with a number that is non-zero: The entire column is divided or multiplied by the number used before. It is depicted as Ci → k Ci, which shows that each component of the column has been scaled up by the factor ‘k’.
5. Adding a column to another one, scaling by a number that is non-zero: Every component from the column is changed by the resultant number found by the addition of the scaled component of a different column. It is depicted as Ci → Ci + k Cj.

A set of two matrices are known as column equivalents only if one of the matrices can be found from the other by conducting any of these three elementary transformations for columns.

 Conclusion

The inverse of a matrix is a rectangular structure of components that are arranged in rows and columns. Analysis of inverse of a matrix by elementary operations can be performed on square matrices.