Elementary Transformation of Matrices: Elementary Row Transformation

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We have already seen that two matrices are equal when they are of the same order and their corresponding elements are equal. That is, if and are two matrices such that then:

  • and i.e. the orders of the two matrices must be same

  • For every value of i and j,

But there is another way to compare matrices. In this case, the matrices are equivalent to each other. For a matrix to be equivalent to a matrix, i.e. , the following two conditions must be satisfied:

  • and again, the orders of the two matrices must be same

  • P should get transformed to Q using the elementary transformation and vice-versa.

Elementary transformation of matrices is hence very important. It is used to find equivalent matrices and also to find the inverse of a matrix. Elementary transformation basically is playing with the rows and columns of a matrix. Let us now go ahead and learn how to transform matrices.

Elementary Transformation of Matrices

Elementary Transformation of Matrices

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Elementary Row Transformation

Only the rows of the matrices is transformed, and no changes are made in the columns. These row operations are executed according to certain set of rules which make sure that the transformed matrix is equivalent to the original matrix. These rules are:

  • Any two rows are interchangeable. The interchange of and rows is represented as:

For example: If , applying

We get Here .

All the elements of any row can be multiplied to any non-zero number. The multiplication of row to a non-zero number k, is represented as:

For example: If , applying

, we get , Again .+

  • All the elements of a row can be added to corresponding elements of another row multiplied by any non-zero constant. If the elements of row are being added to the elements of row (multiplied by a non-zero number k), then it is represented as:

An important thing to note here is that the change will only reflect in the elements of row. The elements of row will remain same.

For example: If , applying

We get

Here, .

It’s easy to figure out here that if , it just reduces to adding the elements of row to corresponding elements of row. Also, if , it means subtracting the elements of row from corresponding elements of the row.

For example: If , applying

We get ,

Those were the three elementary row transformations/operations. The elementary column operations are exactly the same operations done on the columns.

Elementary Column Operation

Those three operations for rows, if applied to columns in the same way, we get elementary column operation. As we have already discussed row transformation in detail, we will briefly discuss column transformation. The rules for elementary column transformation are:

  • Any two columns are interchangeable i.e

For example: If , applying ,

We get

Here .

All the elements of any column can be multiplied to any non-zero number i.e.

For example: If applying , we get

Again

All the elements of a column can be added to corresponding elements of another column multiplied by any non-zero constant i.e.

For Example: If applying

, we get

Here

Those were the elementary transformation techniques for matrices. As mentioned before, these are very important and are used to find the inverse of a matrix.

Question:

Show that matrices A and B are row equivalent if

and

Solution:

Consider the matrix A. Apply row transformation such that

Applying row transformations to the first row,

So, matrix A will be equal to

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

So, the element of second row in A will be given as follow:

So, matrix A will be equal to

Retain and apply row transformation to such that

So, the matrix A will be equal to matrix B.

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