Elementary Transformation of a matrix : The following operation on a matrix are called…

Elementary Transformation of a matrix :
The following operation on a matrix are called elementary operations (transformations)
1 . The interchange of any two rows (or columns)
2 . The multiplication of the elements of any row (or column) by any non zero number
The addition to the elements of any row (or column) the corresponding elements of any other row (or column) the corresponding elements of any other row (or column) multiplied by any number
Echelon form of matrix :
A matrix A is said to be in echelon from if
(i) every row of A which has all its elements 0, occurs below row, which has a non-zero elements
(ii) The first non zero element in each non-zero row is 1 .
(iii) the number of zeros before the first non zero elements in a row is less than the number of such zeroes in the next row.
[A row of matrix is said to be a zero row if all its elements are zero]
Note : Rank of a matrix does not change by application of any elementary operations.
For example \[\left[\begin{array}{lll} 1 & 1 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array}\right],\left[\begin{array}{llll} 1 & 1 & 3 & 6 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]are echelon forms
The number of non-zero rows in the echelon form of a matrix is defined as its RANK.
For example we can reduce the matrix \[A=\left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 7 \\ 3 & 6 & 10 \end{array}\right]\]into echelon form using following elementary row transformation.
(i) R₂ \(\text { → }\) R₂ – 2R₁ and R₃ \(\text { → }\) R₃ – 3 R₁ \[\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{array}\right]\]
(ii) R₂ → R₂ – 2 R₁ \[\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right]\]
This is the echelon form of matrix A
Number of non zero rows in the echelon form = 2
\(\text { ⇒ }\) Rank of the matrix A is 2
Rank of the matrix \[\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 & -1 \\ 3 & 1 & 1 \end{array}\right]\] is