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Elementary Operation (Transformation) of a Matrix

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There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.  
(i) The interchange of any two rows or two columns. Symbolically the interchange of ith and jth rows is denoted by `R_i ↔ R_j` and interchange of `i^(th)` and` j^(th)` column is denoted by `C_i ↔ C_j`.

For example, applying `R_1 ↔ R_2` to
A = `[(1,2,1),(-1,sqrt3,1),(5,6,7)]` ,

we get  `[(-1,sqrt3,1),(1,2,1),(5,6,7)]`

(ii) 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`. 
The corresponding column  operation is denoted by `C_i → kC_i`
For example, applying `C_3 -> 1/7 C_3` , to B = `[(1,2,1),(-1,sqrt3,1)]`, we get `[(1,2,1/7),(-1,sqrt3,1/7)]`

(iii) The addition to 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 to the elements of `i^(th)` row, the corresponding elements of `j_(th)` row multiplied by k is denoted by `R_i → R_i + kR_j`.
The corresponding column operation is denoted by`C_i → C_i + kC_j.`
For example, applying  `R_2 → R_2 – 2R_1`, to C =` [(1,2),(2,-1)]`, we get `[(1,2),(0,-5)]`
Video link : https://youtu.be/tvGi0suZW2U

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