Elementary Transformation of Matrix

Introduction:

A matrix is the rectangular arrangement of numbers, symbols, or characters representing a set of data in any system. The elements of a matrix are arranged in rows and columns. Order of a matrix is the representation of its rows and columns number in the form m x n where m is the number of rows and n is the number of columns. Two matrices are said to be equal if they have the same order and their elements are identical. There is a difference between the terms ‘equal matrices’ and ‘equivalent matrices’. Equivalence of two matrices is denoted using the symbol ‘~’. Two matrices are said to be equivalent if one matrix can be modified through the elementary transformation of the matrix to obtain the other matrix. 


What is Elementary Transformation of Matrix:

Elementary transformations are those operations performed on rows and columns of the matrices to transform it into a different form so that the calculations become simpler. The concept of ‘What is Elementary transformations’ are used in the gaussian method of solving linear equations, determining the echelon form of a matrix and other operations involving matrix representation of a system of equations. It is also used in finding the inverse of the matrices, determinants of the matrices and solving a system of linear equations. To perform elementary transformations between any two matrices, the order of the two matrices must be the same. 


Elementary Row Transformations:

Row transformations are performed only on the basis of a few sets of rules. An individual cannot perform any other kind of row operation apart from the below-stated rules. There are three kinds of elementary row transformations. 

  1. Interchanging the rows within the matrix: In this operation, the entire row in a matrix is swapped with another row. It is symbolically represented as Ri ↔ Rj, where i and j are two different row numbers.

  2. Scaling the entire row with a non zero number: The entire row is multiplied with the same non zero number. It is symbolically represented as Ri → k Ri which indicates that each element of the row is scaled by a factor ‘k’.

  3. Add one row to another row multiplied by a non zero number: Each element of a row is replaced by a number obtained by adding it to the scaled element of another row. It is symbolically represented as Ri → Ri + k Rj. 

Two matrices are said to be row equivalent if and only if one matrix can be obtained from the other by performing any of the above elementary row transformations. 


Example for Row Equivalent Matrices

1. Show that matrices A and B are row equivalent if 

A = \[\begin{bmatrix} 1 & -1  & 0 \\2  &1  &1\end{bmatrix}\] and B = \[\begin{bmatrix} 3 & 0  & 1 \\0  &3  &1\end{bmatrix}\]  

Solution:

Consider the matrix A. Apply row transformation such that R1 → R1 + R2

Applying row transformations to the first row, A11 = 1 + 2, A12 = -1 + 1 and A13 = 0 + 1

So matrix A will be equal to  

\[\begin{bmatrix} 3 & 0  & 1 \\2  &1  &1\end{bmatrix}\]

Now let us retain the first row and apply row transformation to the second row such that 

R2 → 3 R2 - R1

So the elements of second row in A will be given as follows:

A21 = 2 x 3 - 3 = 3

A22 = 1 x 3 - 0 = 3

A23 = 1 x 3 - 1 = 2

So matrix A will be equal to 

\[\begin{bmatrix} 3 & 0  & 1 \\3  &3  &2\end{bmatrix}\]

Retain R1 and apply row transformation to R2 such  that R2 → R2 - R1.

A21 = 3 - 3 = 0

A22 = 3 - 0 = 3

A23 = 2 - 1 = 1

So the matrix A will be equal to matrix B.

\[\begin{bmatrix} 3 & 0  & 1 \\0  &3  &1\end{bmatrix}\]

From this, we can conclude that A and B are row equivalent matrices. 


Elementary Column Transformations:

There are also a few sets of rules to be followed while performing column transformations. There are three different forms of elementary column transformations. No other column transformations are allowed apart from these three. 

  1. Interchanging the columns within the matrix: In this operation, the entire column in a matrix is swapped with another column. It is symbolically represented as  Ci ↔ Cj, where i and j are two different column numbers.

  2. Multiplying the entire column with a non zero number: The entire column is multiplied or divided by the same non zero number. It is symbolically represented as Ci → k Ci which indicates that each element of the column is multiplied by a scaling factor ‘k’.

  3. Add one column to another column scaled by a non zero number: Each element of a column is replaced by a number obtained by adding it to the scaled element of another column. It is symbolically represented as Ci → Ci + k Cj. 

Two matrices are said to be column equivalent if and only if one matrix can be obtained from the other by performing any of the above elementary column transformations.


Fun Facts:

  • Equal matrices have the same order and the same elements.

  • Equivalent matrices are the matrices with the same order and similar elements. Two matrices are said to be equivalent if one matrix can be obtained from the other using the idea of ‘What is Elementary transformation’.

FAQ (Frequently Asked Questions)

1. How is the Rank of a Matrix Determined using Elementary Transformations?

Ans. The number of linearly independent row vectors or linearly independent column vectors in a matrix is called the rank of a matrix. By applying row transformation or column transformation, the given matrix is transformed into its echelon form. Once the matrix is converted into its echelon form, count the number of non zero rows or non zero columns. The number of non zero rows or the non zero columns is called the rank of the matrix. The rank of the matrix can be determined either by applying row transformations or column transformations to derive the echelon matrix.

2. Explain Briefly, the Matrix Notation.

Ans. 

  • Matrices are represented by uppercase bold English fonts. 

  • The order of the matrix is denoted as m x n where’ is the number of rows and ‘n’ is the number of columns.

  • The elements of the matrices are represented in the form of ‘ij’ where ‘i’ is the row number which may vary from 1 to m and ‘j’ is the column number which may vary from 1 to n. Example: An element in the 5th row and 3rd column of matrix A is represented as A53.

  • The rows and columns of any matrix is generally enclosed within a pair of parentheses or square brackets.