Elementary Transformations

A matrix is a rectangular pattern of numbers, symbols, or characters that represent a set of data in any system. A matrix’s elements are organised in columns and rows. The representation of a matrix’s rows & column numbers in the form m x n, where m is the number of rows & n is the number of columns, is called its order. When two matrices have the same order and same elements, they are said to be equivalent. The terms ‘equal matrices’ and ‘equivalent matrices’ are not interchangeable. The sign “~” is used to express the equivalence of two matrices. Two matrices are equivalent if one matrix can be transformed into the other using the Elementary Transformations.

What is an Elementary Transformation of Matrices

The operations done on the columns and rows of the matrices to convert it into a different arrangement so that the computations become simpler are known as the Elementary Transformations. 

The Gaussian method of solving linear equations, calculating the echelon form of a matrix, and other operations involving matrix representation of the system of equations, all employ the notion of the Elementary Transformations. It can also be used to find the inverse and determinants of matrices and solve a system of linear equations. 

Operational Basics

There are three different types of basic matrix operations.

• Two rows should be switched (or columns).
• Multiply each row (or column) element by a non-zero number.
• Multiply a non-zero value by a row (or column) & add the results to another row (or column).

These operations are known as elementary row operations when performed on rows and elementary column operations when performed on columns.

Row Transformations for Beginners

Only a few sets of rules are used to conduct row conversions. Apart from the restrictions listed here, no other type of row operation can be performed. There are three different types of fundamental row transformations.

• Interchanging rows within a matrix

This operation involves swapping an entire row in a matrix with another row. Ri ↔Rj is the sign for it, with i and j denoting two separate row numbers.

• Multiply row with a non-zero value

Using a non-zero number to scale the entire row: The whole row is multiplied by the same non-zero value. It is symbolically expressed as Ri→k Ri, indicating that a factor of ‘k’ scales each row element.

• Multiply one row by a non-zero value and add to another row

Each row’s element is replaced by a number calculated by multiplying it by another row’s scaled element. Ri→ Ri + k Rj is how it’s written symbolically. If only one of the abovementioned elementary row transformations can generate one matrix from the other, two matrices are said to have been row equivalent.

Simple Column Operation

The elementary column operations are obtained by applying the three-row operations to the columns in the same way. We will now briefly cover the column transformations. The following are the rules:

• Any two columns, like any two rows, can be swapped out.
• Each column element can be multiplied by a non-zero value.
• After adding all the column items, the associated elements of another column can be multiplied by any non-zero constant.

Matrix Operation at a Basic Level

A matrix is a set of numbers arranged in rows and columns. The dimensions of a matrix are the number of columns and rows, which are denoted as m n, where m and n are the number of columns and rows, respectively. Certain basic operations can be executed on a matrix aside from basic mathematical operations. The operations done on columns and rows of a matrix to convert the given matrix into a new form to simplify the calculation are known as A matrix’s basic operations for transformations.

Determinants

A determinant is a specific number in linear algebra that can be calculated from a square matrix. The determinant of a matrix, det(P), |P|, or det P, is indicated by the letters det(P), |P|, or det P. Determinants have several important properties in that they allow us to produce the very same results with varied and simpler entry configurations (elements). 

• The reflection property
• Invariance property
• All-zero property 
• Factor property
• Proportionality/repetitions property
• Sum property
• Scalar multiple property 
• Shifting property
• Triangular property 
• Co-factor matrix property 

These are the main properties of determinants.

Important Characteristics of Determinants 

1. The Property of Reflection:

If the determinant’s rows become columns and the columns become rows, the determinant remains unchanged. This is referred to as the reflection property.

2. The property of being all-zero:

The determinant is 0 if all elements in a row (or column) are zero.

3. Property of Proportionality (Repetition):

The determinant is 0 if all elements of a row (or column) are proportionate (identical) to the elements of another row (or column).

4. Changing the Property:

The determinant’s sign changes when two rows (or columns) are swapped.

5. Multiple Scalar Properties:

When all the elements of a determinant’s row (or column) are multiplied by the same non-zero constant, the determinant is multiplied by the same constant.

6.Shifting property:

The value of the determinant does not change on undergoing the first transformation for rows or columns.

Conclusion

The operations done on the columns and rows of the matrices to convert it into various forms so that the computations become simpler are known as Elementary Transformations. It can also be used to find the inverse and determinants of matrices and solve a system of linear equations.