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 = \[\begin{bmatrix} 3 & 7 & 6\\ 4& 17 & 5\\ 2& 7 & 4 \end{bmatrix}\]
FAQs on Elementary Operation of Matrix
1. What are the elementary operations of a matrix as per the CBSE Class 12 syllabus?
In the context of the CBSE Class 12 Maths syllabus, there are six elementary operations, also known as transformations, which can be applied to a matrix. These are divided into three for rows and three for columns:
- The interchange of any two rows or two columns.
- The multiplication of the elements of any row or column by a non-zero number.
- The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number.
These operations form the basis for methods like finding the inverse of a matrix.
2. What is the main purpose of applying elementary operations to a matrix?
The primary purpose of using elementary operations is to simplify a matrix into a more structured form, such as its row echelon form or reduced row echelon form. This simplification is not just for neatness; it is a fundamental process used to:
- Solve systems of linear equations (Gauss-Jordan elimination).
- Calculate the inverse of a square matrix.
- Determine the rank of a matrix, which indicates the number of linearly independent rows or columns.
3. Can you provide a simple example of an elementary row operation?
Certainly. Let's take a 2x2 matrix A. Suppose we want to apply the operation where we add twice the first row to the second row (R₂ → R₂ + 2R₁).
Original Matrix A:
| 1 2 |
| 3 4 |
Applying the operation R₂ → R₂ + 2R₁:
The first row remains unchanged. The new second row is [3 + 2(1), 4 + 2(2)] = [3 + 2, 4 + 4] = [5, 8].
Resulting Matrix:
| 1 2 |
| 5 8 |
This new matrix is considered equivalent to the original matrix A.
4. What is an elementary matrix and how is it connected to elementary operations?
An elementary matrix is a special type of matrix that is created by performing a single elementary operation on an identity matrix (I). Its importance lies in its connection to matrix multiplication:
- Performing an elementary row operation on a matrix 'A' gives the same result as pre-multiplying 'A' by the corresponding elementary matrix.
- Performing an elementary column operation on 'A' is equivalent to post-multiplying 'A' by the corresponding elementary matrix.
This concept is crucial for theoretical proofs and understanding the structure of matrix transformations.
5. How do elementary row operations help in finding the inverse of a matrix?
Elementary row operations are the core of the Gauss-Jordan elimination method for finding a matrix inverse. The process is as follows:
- Create an augmented matrix by placing the original matrix A next to an identity matrix I, in the form [A | I].
- Apply a sequence of elementary row operations to this entire augmented matrix.
- The goal is to transform the left side (matrix A) into the identity matrix (I).
- The same operations, when applied simultaneously to the right side (matrix I), will automatically transform it into the inverse of A, which is A⁻¹.
When the process is complete, the augmented matrix will be in the form [I | A⁻¹].
6. Is it permissible to mix elementary row and column operations when finding an inverse?
No, this is a critical rule to follow. When finding the inverse of a matrix, you must choose one type of operation and stick with it. You must use either only row operations or only column operations throughout the entire process. Mixing them will lead to an incorrect result because the underlying algebraic properties (A = IA for row operations or A = AI for column operations) would be violated.
7. Why are these transformations called 'elementary'?
These operations are called 'elementary' because they are the most fundamental, reversible, and simple steps you can take to manipulate a matrix without changing the core properties of the system it represents. For instance, when a matrix represents a system of linear equations, each elementary row operation corresponds to a basic manipulation of those equations (like swapping two equations or adding a multiple of one to another) that results in an equivalent system with the same solution set.
8. How are elementary operations on matrices different from operations on determinants?
This is a common point of confusion. While the operations look similar, their effects are vastly different:
- On a Matrix: An elementary operation transforms a matrix into a new, equivalent matrix. The identity of the matrix itself changes.
- On a Determinant: Applying an elementary-style operation changes the value of the determinant in a predictable way. For example, swapping two rows of a matrix is an elementary operation, but doing so to a determinant multiplies its value by -1. Multiplying a row by a scalar 'k' also multiplies the determinant's value by 'k'.
