Elementary Matrix Operations: Row and column operations

Elementary Matrix Operations play a vital role in applications of algebra in mathematics. It helps in solving linear equations and also in finding the inverse of a matrix. Matrix is one of the most powerful tools in mathematics. It's a rectangular array of integers structured in rows and columns, in simple terms. The order or dimension of a matrix is determined by the number of rows and columns. The order of a matrix or array is represented by the formula m x n, where n denotes the number of columns and m is the number of rows. In this article, we will have a closer look at the Let's take a closer look at the definition of Matrix, Elementary operations of Matrix, elementary row operations, elementary column operations etc.

Key Terms: Matrix, Arry of integers, types of matrix, elementary row operations, column operation, integers, row, coloumn

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What is a Matrix?

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A matrix is a rectangular array of integers with rows and columns organised in a grid. A matrix is represented by the array of integers below. An example of a matrix or array is as shown below.

The dimension or order of a matrix refers to the number of rows and columns it contains. By convention, rows are listed first, followed by columns. As a result, we may state that the above matrix's dimension (or order) is m x n, implying that it has m rows and n columns. A matrix has some important elementary matrix operations which comprises of elementary row operations and elementary column operations.

Also Read: Identity Matrix

Matrix

The numbers that appear in the rows and columns of a matrix are referred to as matrix elements. The first column of the first row in the given matrix has the value 21; the second column of the first row has the value 62; and so on.

A matrix

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Determinants Detailed Video Explanation:

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Types of Matrix

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• Transpose Matrix 

Transpose Matrix

The transpose of one matrix is another matrix in which the original matrix's rows are used as columns in the second matrix.

The transpose of matrix A, for example, is easily seen to be A'. The first row of matrix A becomes the first column of A', the second row of A becomes the second column of A', and the third row of A becomes the third column of A'.

• Square Matrix

An n x n matrix, or a matrix with the same number of rows as columns, is known as a square matrix. The different types of square matrices are:

• Symmetric Matrix- A matrix is said to be symmetric if its transpose is the same as its own. 

Also Read: Symmetric Matrix

Symmetric Matrices

• Diagonal Matrix-. A diagonal matrix is a type of symmetric matrix that is unique. It's a symmetric matrix containing off-diagonal components that have zeros. 

Diagonal Matrices

• Scalar Matrix- A scalar matrix is a type of diagonal matrix that is unique. It's a diagonal matrix with components of equal value running diagonally. 

Also Read: Scalar Matrix

Scalar Matrices

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Elementary Operations of Matrix

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The inverse of a matrix is usually found using Elementary Matrix Operation techniques. The following are the two types of elementary matrix operations:

• Primary or Elementary Row Operations: Primary or elementary row operations are operations done on the rows of an array or matrix.
• Primary or Elementary Column Operations: Primary or elementary column operations are the elementary matrix operations performed on its columns.

The three basic elementary matrix operations or elementary operations of a matrix are as follows:

• The interchange of any two rows or columns.
• Multiplication of a row or a column by a non-zero number.
• Multiplication of a row or a column by a non-zero number and adding the result to some other row or column.

Also Read: Singular Matrix

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Elementary Row Operations

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In the case of elementary row operations, the elementary matrix operations are performed only on the rows of a matrix:

Case 1: The Interchange of any Two Rows

In a matrix or array, any two rows can be swapped or switched. When the ith row and the jth row are swapped, the result is Ri ↔ Rj which is an elementary row operation. The matrix A, for example, is shown below.

\(A = \begin{bmatrix} 2&4\\ 1&3 \end{bmatrix}\)

We may achieve the following result by using the elementary row matrix operations R1 ↔ R2

\(A = \begin{bmatrix} 1&3\\ 2&4 \end{bmatrix}\)

Case 2: Multiplication of Row by a Non-zero Number

Any non-zero integer can be used to multiply the elements of any row. Ri ↔ k Ri is the multiplication of the ith row with k (any non-zero value). 

We've presented a matrix A as an example.

A= [2 5 6 3]

So, if we perform the Elementary row operation  R1↔3R1,we obtain: 

A= [6 15 18 9]

Case 3: Multiplication of Row by a Non-zero Number and Add the Result to the Other Row 

After multiplying the matrix with any non-zero value, we may add the elements of any row with the corresponding elements of another row. The sum of the elements of an ith row and the elements of the jth row, multiplied by k (any non-zero number), is symbolically expressed as Ri ↔ Ri + k Rj. 

We've presented a matrix A as an example.

\(A = \begin{bmatrix} 2&4\\ 1&3 \end{bmatrix}\)

We acquire  by using the elementary rpw operation R2↔R2+2R1

\(A = \begin{bmatrix} 2&4\\ 7&15 \end{bmatrix}\)

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Elementary Column Operations

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In the case of elementary column operations, the elementary matrix operations are performed only on the columns of a matrix:

Case 1: The Interchange of any Two Columns

In a matrix or array, any two columns can be swapped or switched. When the ith column and the jth column are swapped, Ci ↔ Cj denotes the substitution. Suppose we have:

\(A = \begin{bmatrix} 2&4\\ 1&3 \end{bmatrix}\)

We may achieve the following result by using the elementary column matrix operations C1 ↔ C2

\(A = \begin{bmatrix} 4&2\\ 3&1 \end{bmatrix}\)

Case 2: Multiplication of a Column by a Non-zero Number

Any non-zero integer can be used to multiply the elements of any column. Ci ↔ k Ci is the multiplication of the ith column with k (any non-zero value). 

We've presented a matrix A as an example.

\(A = \begin{bmatrix} 2&4\\ 1&3 \end{bmatrix}\)

So, if we perform the Elementary column operation  C1↔ 3C1,we obtain: 

\(A = \begin{bmatrix} 6&4\\ 3&3 \end{bmatrix}\)

Case 3: Multiplication of a Column by a Non-zero Number and Add the Result to the Other Column

After multiplying the matrix with any non-zero value, we may add the elements of any column with the corresponding elements of another column. The sum of the elements of an ith column and the elements of the jth column, multiplied by k (any non-zero number), is symbolically expressed as Ci ↔ Ci + k Cj. 

We've presented a matrix A as an example.

\(A = \begin{bmatrix} 2&4\\ 1&3 \end{bmatrix}\)

We acquire  by using the elementary column operation C2↔C2+2C1

\(A = \begin{bmatrix} 2&8\\ 7&5 \end{bmatrix}\)

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Things to Remember

• A matrix is a collection of integers organised into rows and columns. 
• The dimensions of a matrix are the number of rows and columns, which are denoted as m x n, where m and n are the number of rows and columns, respectively. 
• There are certain fundamental operations that can be done on a matrix known as the  elementary matrix operations which are elementary row operations and elementary column operations.
• A matrix's three fundamental elementary operations or transformations are as follows:

1. Any two rows or columns can be swapped.
2. Multiplication of a non-zero value by a row or column.
3. Add the result to the other row or column after multiplying the row or column by a non-zero value.

Also Read: Adjacency Matrix