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Elementary Row Operations
There are many applications of elementary row operations. They can be used to solve a system of equations in an easy way, they can be used to find the rank of a matrix etc. The elementary row transformations are also used to find the inverse of a matrix A without using any formula like A^{1 }= (adj A) / (det A).
Let us see how to apply inverse row operations for performing multiple things in an easy way.
What are Elementary Row Operations?
While applying the elementary row operations, we usually represent the first row by R₁, the second row by R₂, and so on. There are primarily three types of elementary row operations:
 Interchanging two rows.
For example, interchanging the first and second rows is shown by R₁ ↔ R₂.  Multiplying/dividing a row by a scalar.
For example, if the first row (all elements of the first row) is multiplied by some scalar, say 3, it is shown as R₁ → 3R₁.  Multiplying/dividing a row by some scalar and adding/subtracting to the corresponding elements of another row.
For example, if the first row is multiplied by 3, and added to the second row, we can write it either as R₁ → 3R₁ + R₂ (or) R₂ → R₂ + 3R₁.
It is a common practice to write the same row on the left side of the arrow and on the very first occurrence of the right side of the arrow.
Elementary Row Operations to Solve a System of Equations
We can solve a system of equations written in matrix form AX = B, by writing the augmented matrix [A B] and applying the elementary row operations on it to convert it into the echelon form (preferably the upper triangular form). Applying all the above three row operations do not alter the augmented matrix as:
 Interchanging two rows is nothing but swapping two equations of the system and this doesn't affect the solution.
 Multiplying a row by some scalar does not alter the augmented matrix, as we always can multiply both sides of an equation by a scalar without affecting the equation.
 Multiplying one row by scalar and adding it to the other row is nothing but multiplying an equation by a scalar and adding it to another equation and we usually do this to solve a system of equations.
This process of applying row operations to solve a system is known as Gauss elimination. We can see an example of applying row transformations to solve a system of equations in the "Elementary Row Operations Examples" section below.
Elementary Row Operations to Find Inverse of a Matrix
To find the inverse of a square matrix A, we usually apply the formula, A^{1} = (adj A) / (det A). But this process is lengthy as it involves many steps like calculating cofactor matrix, adjoint matrix, determinant, etc. To make this process easy, we can apply the elementary row operations. Here are the steps for doing the same.
 Consider the augmented matrix [A  I], where I is the identity matrix that is of the same order as A.
 Apply row transformations to convert the left side matrix A into I.
 Then the right side matrix (that replaces the original matrix I) is nothing but A^{1}.
To see how to find the inverse of a 2x2 matrix and the inverse of a 3x3 matrix using elementary row operations, click on the respective links.
Elementary Row Operations to Find Determinant
Usually, we find the determinant of a matrix by finding the sum of the products of the elements of a row or a column and their corresponding cofactors. But this process is difficult if the terms of the matrix are expressions. But we can apply the elementary row operations to find the determinant easily. But some of the row operations affect the determinant in the following ways:
 Interchanging two rows of a determinant changes its sign.
 Multiplying a row by some scalar multiplies the determinant by the same scalar.
 Multiplying a row by some scalar and adding the result to another row doesn't alter the determinant.
To see how to find the determinant of a matrix by elementary row operations, click here.
Elementary Row Operations to Find Rank of a Matrix
The rank of a matrix is the number of linearly independent rows (or columns) in it. We can apply the elementary row operations on the matrix to find its rank in two ways:
 We can convert it into Echelon form and count the number of nonzero rows in it which gives its rank.
 We can convert it int the normal form \(\left[\begin{array}{ll}
lᵣ & 0 \\ \\
0 & 0
\end{array}\right]\), where Iᵣ is the identity matrix of order r. Then the rank of the matrix = r.
To understand these two methods with examples, click here.
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Elementary Row Operations Examples

Example 1: Perform the following elementary row operations on the matrix A = \(\left[\begin{array}{rrr}
1 & 2 & 1 \\
3 & 2 & 0 \\
4 & 0 & 2
\end{array}\right]\): (a) R₁ ↔ R₂ (b) R₂ → R₂  5R₁.Solution:
(a) R₁ ↔ R₂ means swapping (or interchanging) the first two rows.
Then the result is \(\left[\begin{array}{rrr}
3 & 2 & 0 \\
1 & 2 & 1 \\
4 & 0 & 2
\end{array}\right]\).(b) We have R₁ (the first row) = [1 2 1].
Then 5R₁ = [5 10 5].
R₂  5R₁ = [3 2 0] + [5 10 5] = [2 8 5]
R₂ → R₂  5R₁ means replace R₂ by the row obained by doing R₂  5R₁. Then the resultant matrix is \(\left[\begin{array}{rrr}
1 & 2 & 1 \\
2 & 8 & 5 \\
4 & 0 & 2
\end{array}\right]\).Answer: (a) \(\left[\begin{array}{rrr}
3 & 2 & 0 \\
1 & 2 & 1 \\
4 & 0 & 2
\end{array}\right]\) (b) \(\left[\begin{array}{rrr}
1 & 2 & 1 \\
2 & 8 & 5 \\
4 & 0 & 2
\end{array}\right]\). 
Example 2: Solve the following system of equations using elementary row transformations: 2x  y + 3z = 8, x + 2y + z = 4, and 3x + y  4z = 0.
Solution:
The matrix equation of the given system is:
\(\left[\begin{array}{ccc}
2 & 1 & 3 \\
1 & 2 & 1 \\
3 & 1 & 4
\end{array}\right]\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
8 \\
4 \\
0
\end{array}\right]\)The augmented matrix is,
[A B] = \(\left[\begin{array}{ccc:c}
2 & 1 & 3 & 8 \\
1 & 2 & 1 & 4 \\
3 & 1 & 4 & 0
\end{array}\right]\)Here we convert the last two elements of the first column (1 and 3) to be zero. We use R₁ in this process.
Apply R₂ → 2R₂ + R₁ and R₃ → 2R₃  3R₁, we get:
= \(\left[\begin{array}{ccc:c}
2 & 1 & 3 & 8 \\
0 & 3 & 5 & 16 \\
0 & 5 & 17 & 24
\end{array}\right]\)We convert the last element of the second column (5) to be a zero. We use R₂ in this process.
Now, apply R₃ → 3R₃  5R₂,
= \(\left[\begin{array}{ccc:c}
2 & 1 & 3 & 8 \\
0 & 3 & 5 & 16 \\
0 & 0 & 76 & 152
\end{array}\right]\)Now, we expand the above matrix as equations:
2x  y + 3z = 8 ... (1)
3y + 5z = 16 ... (2)
76z = 152 ... (3)
From (3), z = (152) / (76) = 2.
From (2), 3y + 5(2) = 16 ⇒ 3y = 6 ⇒ y = 2.
From (1), 2x  2 + 3 (2) = 8 ⇒ 2x = 4 ⇒ x = 2.
Answer: (x, y, z) = (2, 2, 2).

Example 3: Find the inverse of the matrix A = \(\left[\begin{array}{ccc}
2 & 1 & 3 \\
0 & 1 & 1 \\
1 & 2 & 0
\end{array}\right]\) using elementary row operations.Solution:
Consider the augmented matrix formed by A and the identity matrix I.
[A  I] = \(\left[\begin{array}{ccc:ccc}
1 & 0 & 0 & 2 & 1 & 3 \\
0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 2 & 0
\end{array}\right]\)We will convert the right side matrix as the identity matrix.
Apply R₃ → 2R₃ + R₁,
= \(\left[\begin{array}{ccc:ccc}
1 & 0 & 0 & 2 & 1 & 3 \\
0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 2 & 0 & 5 & 3
\end{array}\right]\)Now apply R₁ → R₁ + R₂ and R₃ → R₃ + 5R₂,
\(\left[\begin{array}{ccc:ccc}
1 & 1 & 0 & 2 & 0 & 4 \\
0 & 1 & 0 & 0 & 1 & 1 \\
1 & 5 & 2 & 0 & 0 & 8
\end{array}\right]\)Apply R₁ → 2R₁  R₃ and R₂ → 8R₂  R₃,
\(\left[\begin{array}{ccc:ccc}
1 & 3 & 2 & 4 & 0 & 0 \\
1 & 3 & 2 & 0 & 8 & 0 \\
1 & 5 & 2 & 0 & 0 & 8
\end{array}\right]\)Now divide R₁ by 4, R₂ by 8, and R₃ by 8:
\(\left[\begin{array}{ccc:ccc}
1 / 4 & +3 / 4 & +2 / 4 & 1 &0 &0 \\
+1 / 8 & 3 / 8 & 2 / 8 & 0 &1 & 0 \\
1 / 8 & 5 / 8 & 2 / 8 & 0 & 1 & 0
\end{array}\right] .\)Now, the right side matrix got converted into I. Hence, the left side matrix is A^{1}.
Answer: A^{1 }=^{ }\(\left[\begin{array}{ccc:ccc}
1 / 4 & 3 / 4 & 2 / 4 \\
1 / 8 & 3 / 8 & 2 / 8 \\
1 / 8 & 5 / 8 & 2 / 8
\end{array}\right] .\)
FAQs on Elementary Row Operations
How to Apply Elementary Row Operations?
We can apply three types of elementary row operations:
 We can interchange two rows.
 We can multiply/divide any row(s) by a number.
 We can multiply/divide a row by some number and add/subtract it to another row.
How to Know Which Elementary Row Operations Have to be Applied to Solve a System of Equations?
While solving the system of equations (3x3) AX = B using augmented matrix [A B]:
 First aim at making the last two elements of the first column as zeros. Use row 1 for this process.
 Then aim at making the last element of the second column as zero. Use row 2 for this process.
Then the matrix gets converted into the upper triangular matrix. Expand it back into equations and solve them easily.
Do Elementary Row Operations Affect the Determinant?
Some row operations affect the determinant. Swapping two rows changes the sign of the determinant. Multiplying a row by some number multiplies the actual determinant also by the same factor. But multiplying a row by some number and adding it to the other row does not affect the determinant.
How to Apply Elementary Row Transformations to Find the Rank of a Matrix?
To find the rank of a matrix, convert it into echelon form (upper triangular matrix or lower triangular matrix) by applying elementary row operations. Finally, count the number of nonzero rows in it. This gives the rank of the matrix.
Do Elementary Row Operations Affect the System of Equations?
No, any type of elementary row operation does not affect the system of equations. Thus, applying row transformations is an effective way of solving a system of equations.
Explain the Process of Using Elementary Row Operations to Find Inverse of a Matrix.
To find the inverse of a square matrix A:
 Take the matrix [A  I], where I is the identity matrix of same order as that of A.
 Apply elementary row transformations to convert it to the form [I  a square matrix]
 "A square matrix" in the above step gives A^{1}.
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