Elementary Operations! Let’s get a deeper understanding of what they actually are and how are they useful. Elementary operations for matrices play a crucial role in finding the inverse or solving linear systems. They may also be used for other calculations. The matrix on which elementary operations can be performed is known as an elementary matrix.

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## What is an Elementary Matrix?

An elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation (or column operation). Now, let’s consider a matrix given below,

Its rows are

Its columns are

Let us consider the matrix transpose of *A*

Its rows are

*Learn Types of Matrices here.*

**Browse more Topics under Matrices**

- Matrix
- Types of Matrices
- Addition of Matrices
- Scalar Multiplication of Matrices
- Symmetric and Skew-Symmetric Matrices
- Multiplication of Matrices
- Transpose of a Matrix
- Invertible Matrices

## Elementary Matrix Operations

- Interchange two rows or columns.
- Multiply a row or a column with a non-zero number.
- Add a row or a column to another one multiplied by a number.

### 1. The interchange of any two rows or two columns

Symbolically the interchange of the i^{th} and j^{th} rows is denoted by R_{i} ↔ R_{j} and interchange of the i^{th} and j^{th} column is denoted by C_{i} ↔ C_{j}. For example, applying R_{1} ↔ R_{2 }to $$A=\begin{bmatrix} 3 & -4 & 5\\ 1 & 1 & 2\\4 & 5 & 7\end{bmatrix}$$ gives $$A=\begin{bmatrix} 1 & 1 & 2\\3 & -4 & 5\\4 & 5 & 7\end{bmatrix}$$

### 2. The multiplication of the elements of any row or column by a non zero number

Symbolically, the multiplication of each element of the i^{th} row by k, where k ≠ 0 is denoted by R_{i }→ kR_{i.} For example, applying R_{1} → 1 /2 R_{1 }to $$A=\begin{bmatrix} 1 & 1 & 2\\ 3 & -4 & 5\\4 & 5 & 7\end{bmatrix}$$ gives $$A=\begin{bmatrix} 1/2 & 1/2 & 1\\3 & -4 & 5\\4 & 5 & 7\end{bmatrix}$$

### 3. The addition of the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number.

Symbolically, the addition of the elements of the i^{th} row, the corresponding elements of j^{th} row multiplied by k is denoted by R_{i} → R_{i} + kR_{j}. For example, applying R_{2} → R_{2} –2 R_{1}, to $$A=\begin{bmatrix} 1 & 1 & 2\\ 3 & -4 & 5\\4 & 5 & 7\end{bmatrix}$$ gives $$A=\begin{bmatrix} 1 & 1 & 2\\1 & -6 & 1\\4 & 5 & 7\end{bmatrix}$$

## Row Equivalence

Two matrices are row equivalent if and only if one may be obtained from the other one via elementary row operations. For example, show that the two matrices are row equivalent.

Answer: We start with *A*. If we keep the second row and add the first to the second, we get

We keep the first row. Then we subtract the first row from the second one multiplied by 3. Hence, we get

We keep the first row and subtract the first row from the second one. Hence, we get

*[source: sosmath]*

which is the matrix *B*. Therefore *A* and *B* are row equivalent. The column equivalence also implies the same rule.

## Solved Examples For You

**Question 1: If you switch the first row with the fourth row, what will the new first row be?**

**$$A=\begin{bmatrix} 3 & 4 & 2 & 11\\9 & 1 & 0 & 0\\0 & 1 & 0 & 2\\0 & 0 & 6 & 1\end{bmatrix}$$**

**3, 4, 2, 11****9, 1, 0, 0****0, 1, 0, 2****0, 0, 6, 1**

**Answer :** Option (D) 0, 0, 6, 1

**Question 2: $$A=\begin{bmatrix} 1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}$$**

**B is matrix obtained by subtracting 4 times 1st rowfrom 2nd row of A. C is matrix obtained by subtracting 7 times 1 ^{st} row from 3^{rd} row, then C is**

**$$\begin{bmatrix} 1 & 2 & 3\\0 & -3 & -6\\0 & -3 & -12\end{bmatrix}$$****$$\begin{bmatrix} 1 & 2 & 3\\0 & 1 & 2\\3 & 4 & 5\end{bmatrix}$$****neither (A) nor (B)****cannot be calculated**

**Answer :** Option (A)

$$C=\begin{bmatrix} 1 & 2 & 3\\0 & -3 & -6\\0 & -3 & -12\end{bmatrix}$$

**Question 3: Define the elementary operation?**

**Answer:** The elementary operations are generally the operations in the elementary arithmetic. These are; the addition, multiplication, subtraction, and division. Moreover, the elementary row operations or the elementary column operations.

**Question 4: What is the elementary row operation on a matrix?**

**Answer:** The ‘basic operation’ on the numerical values and inputs is the addition, subtraction, multiplication, and division. For matrix, there are 3 basic row operations, this means there are 3 techniques that we can do with the rows of the matrix.

**Question 5: What is the rank when it comes to a matrix?**

**Answer:** The rank of a matrix refers to (A) ‘the highest possible number of the linearly independent column vectors in a matrix’ or (B) that says ‘the greatest possible number of the linearly self-determining row vectors in a matrix.

**Question 6: What is the product of an elementary matrix?**

**Answer:** An elementary matrix basically refers to a matrix that we can achieve from the identity matrix by a single elementary row operation. On multiplying the matrix ‘A’ by the elementary matrix ‘E’ it results in ‘A’ to go through the elementary row operation symbolized by ‘E’.

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