Elementary Transformation of Matrix – Introduction, Definition

What is Elementary Transformation of the Matrix?

Elementary transformation of the matrix is the transformation of a matrix in which the matrix is multiplied by a scalar. The scalar can be either a real number or an imaginary number. Elementary Transformation of Matrix – Introduction Definition.

Elementary Row Transformations

There are three elementary row transformations that can be performed on a matrix A:

1. Rotation: A rotates its rows by a certain angle θ.
2. Reflection: A reflects its rows across a certain line y = x.
3. Transposition: A transposes its rows by swapping the first row with the last row.

Example for Row Equivalent Matrices

The row equivalent matrices are:

A =

B =

C =

D =

The matrices are row equivalent because they have the same number of rows and the same number of columns. The corresponding elements in each matrix are also equal.

Elementary Column Transformations

There are three basic types of column transformations:

1. Rotations
2. Reflections
3. Shear

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

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

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

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.

From this, we can conclude that A and B are row equivalent matrices. Elementary Transformation of Matrix – Introduction .

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. Elementary Transformation of Matrix – Introduction .

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’.