Types of elementary matrix operations with examples and properties
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 Explained for Students
1. What are elementary operations of a matrix?
The elementary operations of a matrix are the three basic row or column operations used to simplify a matrix or solve systems of equations. These operations are:
- Row interchange (Ri ↔ Rj) – swapping two rows.
- Row scaling (Ri → kRi) – multiplying a row by a non-zero constant k.
- Row replacement (Ri → Ri + kRj) – adding a multiple of one row to another row.
2. What are the three elementary row operations?
The three elementary row operations are row interchange, row scaling, and row replacement. Specifically:
- Ri ↔ Rj (Interchange two rows)
- Ri → kRi, where k ≠ 0 (Multiply a row by a non-zero scalar)
- Ri → Ri + kRj (Add a multiple of one row to another)
3. What is an example of an elementary row operation?
An example of an elementary row operation is replacing a row with the sum of itself and a multiple of another row. For example, given the matrix:
- [[1, 2], [3, 4]]
- New R2 = [3, 4] − 3[1, 2] = [3−3, 4−6] = [0, −2]
4. Why are elementary operations important in matrices?
Elementary operations are important because they help simplify matrices without changing the solution of a system of linear equations. They are used to:
- Convert a matrix into row echelon form (REF)
- Find the rank of a matrix
- Compute the inverse of a matrix
- Solve linear systems using Gaussian elimination
5. What is the difference between row operations and column operations?
The main difference is that row operations act on rows while column operations act on columns of a matrix. Row operations are mainly used to solve linear systems and find inverses, whereas column operations are often used in determinant evaluation and theoretical proofs. Both follow the same three types: interchange, scaling, and replacement.
6. Do elementary row operations change the determinant?
Yes, elementary row operations can change the determinant of a matrix depending on the operation performed. The effects are:
- Row interchange → determinant changes sign.
- Row scaling by k → determinant is multiplied by k.
- Row replacement (Ri → Ri + kRj) → determinant remains unchanged.
7. How do you use elementary operations to find the inverse of a matrix?
To find the inverse of a matrix, apply elementary row operations to transform [A | I] into [I | A⁻¹]. The steps are:
- Write the augmented matrix [A | I].
- Use row operations to convert A into the identity matrix I.
- The right side becomes A⁻¹.
8. What is row echelon form in matrix operations?
A matrix is in row echelon form (REF) if it has leading 1s that move to the right as you go down the rows and all entries below each leading 1 are zero. The conditions are:
- All zero rows are at the bottom.
- Each leading entry is to the right of the one above it.
- Entries below each leading entry are zero.
9. Can elementary row operations change the rank of a matrix?
No, elementary row operations do not change the rank of a matrix. The rank depends on the number of linearly independent rows or columns, and row operations preserve linear dependence relations. Therefore, row-reduced forms are commonly used to determine matrix rank.
10. What are common mistakes when performing elementary operations on matrices?
Common mistakes in elementary matrix operations include calculation errors and incorrect application of operations. Typical errors are:
- Multiplying a row by zero (not allowed in scaling).
- Forgetting to apply the operation to every element of the row.
- Changing two rows unintentionally instead of one.
- Incorrect arithmetic during Gaussian elimination.
