Elementary Operation of Matrix

Matrix is one of the most powerful tools in mathematics. In simple words, it is a rectangular array of numbers organized in rows and columns. The number of rows and columns in a matrix determines its order or dimension. The general representation of the order of a matrix or array is m X n, where n represents the number of columns, while m represents the number of rows. The following is an example of a matrix or array.

\[\begin{bmatrix} 1 & 3 &2 \\ 6 &2 &7 \\ 3 &4 & 7 \end{bmatrix}\]

The above matrix has three rows and three columns. Hence, the order of this array is 3 X 3. There are many operations that you can perform on a matrix, which are known as transformations. Now, let’s look at the Elementary Operation of Matrix in detail in the article below.

Types of Elementary Operations

Elementary operations are mostly used to find the inverse of the matrix. The two types of matrix elementary operations are:

  • Elementary Row Operations: Elementary operations performed on the rows of the array or matrix are known as primary or elementary row operations.

  • Elementary Column Operations: The elementary matrix operations performed on its columns are known as primary or elementary column operations. 

Elementary Operation of Matrix Rules

The following are the rules of the elementary operations of the matrix.

  • Any two columns or rows in a matrix or array can be interchanged or exchanged. When we interchange ith row with jth row, then it is written as Ri ↔ Rj. The exchanging of the ith column with the jth column can be written as Ci ↔ Cj.

For example, below is the matrix A

A = \[\begin{bmatrix} 1 & 2\\ 5 & 3 \end{bmatrix}\]

By applying the elementary matrix operations R1 ↔ R2, we get

A = \[\begin{bmatrix} 5 & 3\\ 1 & 2 \end{bmatrix}\]

We can multiply the elements of any row (or column) by any non-zero number. We can write the multiplication of ith row with k (any non-zero number) as Ri ↔ k Ri. If we multiply the jth column with k, we can denote it symbolically as Cj ↔ k Cj

For example, we have given a matrix A

A = \[\begin{bmatrix} 2 & 5\\ 6 & 3 \end{bmatrix}\]

If we apply the elementary operation R1 ↔ 3 R1, then we get

A = \[\begin{bmatrix} 6 & 15\\ 6 & 3 \end{bmatrix}\]

  • We can add the elements of any row (or column) with the corresponding elements of another row (or column) of the matrix after multiplying it with any non-zero number. The addition of the elements of an ith row with the jth row, which is multiplied by k (any non-zero number), can be symbolically denoted as Ri ↔ Ri + k Rj. Similarly, we can add the elements of the ith column to the jth column, which is multiplied by k that we can symbolically write as Ci ↔ Ci + k Cj.

For example, we have given a matrix A

A = \[\begin{bmatrix} 2 & 3\\ 6 & 2 \end{bmatrix}\]

By applying the elementary operation R2 ↔ R2 + 2R1, we get

A = \[\begin{bmatrix} 4 & 3\\ 14 & 8 \end{bmatrix}\]


Solved Examples

In this section of this article, we have given some matrix elementary operations examples that help you to understand the topic more clearly. 

Example 1: Apply the elementary operation C2 ↔ C1 on a 3 X 3 matrix A. Given that .

A = \[\begin{bmatrix} 3 & 7 & 2\\ 4& 8 & 3\\ 6& 9 & 1 \end{bmatrix}\]

Answer: We have given that

A = \[\begin{bmatrix} 3 & 7 & 2\\ 4& 8 & 3\\ 6& 9 & 1 \end{bmatrix}\]

Now, we have to apply the elementary matrix operation C2 ↔ C1. It means we have to interchange the column 2 with column 1. After using this column operation C2 ↔ C1 on A, we get

A = \[\begin{bmatrix} 4 & 8 & 3\\ 3& 7 & 2\\ 6& 9 & 1 \end{bmatrix}\]


Example 2: Apply the elementary operation R2 ↔ 1/2R2 on matrix A. Given that 

A = \[\begin{bmatrix} 2 & 3 & 8\\ 6& 2 & 10\\ 9& 6 & 5 \end{bmatrix}\].

Answer: Given that

A = \[\begin{bmatrix} 2 & 3 & 8\\ 6& 2 & 10\\ 9& 6 & 5 \end{bmatrix}\]

Now, we have to apply the elementary operation R2 ↔ 1/2R2 on A. It means we have to multiply ½ with every element present in the second row of A, i.e., A21 ↔ ½ A21, A22 ↔ ½ A22, A23 ↔ ½ A23

Hence, A21 will become ½ X 6= three after applying the given elementary operation. Similarly, A22 will become 1, and A23 will become 5.

The matrix obtained after applying the given elementary operation is.   

A = \[\begin{bmatrix} 2 & 3 & 8\\ 3& 1 & 5\\ 9& 6 & 5 \end{bmatrix}\]


Example 3:  Find the matrix obtained after applying the elementary operation C2 ↔ C2 + 2C1 on the below array or matrix..

A = \[\begin{bmatrix} 3 & 1 & 6\\ 4& 9 & 5\\ 2& 3 & 4 \end{bmatrix}\].

Answer: We have given that 

A = \[\begin{bmatrix} 3 & 1 & 6\\ 4& 9 & 5\\ 2& 3 & 4 \end{bmatrix}\]

Now, we have to apply the elementary operation of matrix C2 ↔ C2 + 2C1 to A. It means that every second column element will become the addition of its given elements with corresponding elements of the first column after multiplying with 2. Hence, A12 ↔ A12 + 2A11, A22 ↔ A22 + 2A21and A32 ↔ A32 + 2A31

Therefore, A12 will become 1 + 2 X 3= 7

Similarly, A22 will become 9 + 2 X 4= 17 and A32 will become 3 + 2 X 2= 7

The final matrix obtained after applying the given elementary operation is.

A = \[\begin{bmatrix} 3 & 7 & 6\\ 4& 17 & 5\\ 2& 7 & 4 \end{bmatrix}\]

FAQ (Frequently Asked Questions)

Q. What are the Applications of Matrices in Practical Life?

Ans. Matrices are one of the essential topics in maths. They allow us to put a lot of information in just a rectangular array. Hence, it has many applications in physics and engineering. Using a matrix, you don’t have to write significant equations that depend on multi-dimensional quantities separately. Even, many GPS companies are using the matrix system to provide full accuracy to the users. Many IT companies are also using matrices to keep the database of their users. Arrays are also beneficial in plotting graphs and various types of scientific researches. Many economists consider it as the best method to optimize different kinds of problems.

Q. Is it Possible to Determine the Rank of the Matrix Using Elementary Matrix Operations?

 Ans. The number of linearly independent column vectors or row vectors present in it is called the rank of the matrix. We can quickly transform the array into its echelon form by applying column or row elementary operations. After converting the matrix into its echelon form, we have to count the number of non-zero columns or rows. In simple words, the number of non-zero rows or the non-zero columns after applying the elementary matrix operations is called the rank of the matrix. We can either use the elementary row operations or the column operations for getting the echelon form of the matrix.