Question

What are the $3$ elementary row operations?

Answer 1

First, we will need to know the concept of matrix and its order, then we will discuss the row operations.
A matrix is a rectangular entry with elements or variables.
Let us take the Square matrix, same order matrix-like $1 \times 1,2 \times 2,3 \times 3,.....n \times n$.
Here by the row operation method, we will discuss which methods are elementary.

Complete step-by-step solution:
Row operations on the matrix are certain operations that will perform on the matrices.
$1)$ Multiply any row by a constant.
Let us take a matrix \left[ {\begin{array}{*{20}{c}} 1&2 \\\ 2&1 \end{array}} \right] and then multiply with any constant, like with the number $2$ then we get \left[ {\begin{array}{*{20}{c}} 1&2 \\\ 2&1 \end{array}} \right],{R_1} \to 2{R_1} \Rightarrow \left[ {\begin{array}{*{20}{c}} 2&4 \\\ 2&1 \end{array}} \right] (it will multiply the values on row one only)
By this, we are able to simplify the matrix using the row transformation.
$2)$ Exchange any two rows.
Let us take the matrix, \left[ {\begin{array}{*{20}{c}} 1&1&1 \\\ 2&2&2 \\\ 3&3&3 \end{array}} \right] then change the row one to row two represented order, then we get {R_1} \leftrightarrow {R_2},\left[ {\begin{array}{*{20}{c}} 1&1&1 \\\ 2&2&2 \\\ 3&3&3 \end{array}} \right] \Rightarrow \left[ {\begin{array}{*{20}{c}} 2&2&2 \\\ 1&1&1 \\\ 3&3&3 \end{array}} \right] the row one and two are interchanged.
We will use this to find the upper and lower triangle matrices.
$3)$ Add a multiple of one row to another row
Let us take the matrix, \left[ {\begin{array}{*{20}{c}} 1&2 \\\ 2&1 \end{array}} \right] then add the row two elements with any constant like a number $2$with the multiplication of row one, then we get \left[ {\begin{array}{*{20}{c}} 1&2 \\\ 2&1 \end{array}} \right],{R_2} \to {R_2} + 2{R_1} \Rightarrow \left[ {\begin{array}{*{20}{c}} 1&2 \\\ {2 \times 2}&{1 \times 4} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&2 \\\ 4&4 \end{array}} \right]
This is called the row transformation method.
Therefore, the three methods are Multiplying constant, Exchanging rows, and Row transformation method.

Note: A matrix is nonsingular ($\left| A \right| \ne 0$) then we are able to find its inverse form.
If it singular ($\left| A \right| = 0$) then we cannot find its inverse form.
Also, in a matrix nonsingular matrix = invertible matrix (If there is an inverse exists, then the matrix is known as the non-singular because of the determinant non-zero).
If the order of the elements is not equal (not the same size), then it is called a non-square matrix.
$1 \times 2,5 \times 7$
If for example take a $2 \times 2$ matrix which is A = \left[ {\begin{array}{*{20}{c}} 2&1 \\\ 4&3 \end{array}} \right]then we can able to find its ${A^{ - 1}}$
If we see \left| A \right| = \left| {\begin{array}{*{20}{c}} 2&1 \\\ 4&3 \end{array}} \right| = (2 \times 3) - (1 \times 4)
$\Rightarrow 6 - 4$
$\Rightarrow 2$
Thus, the determinant of a matrix $A$ is non zero, that is $\left| A \right| = 2$
Which is a non-singular matrix