Elementary row operations

A matrix is an arrangement of expressions, defined in general terms. The items that are arranged are called as elements. We repeat that it is only an arrangement; thereby a matrix does not suggest any algebraic operation between the elements. Due to this fact a matrix can undergo certain operations with its rows, called as matrix elementary row operations or simply as row operations. The row operations are also called as row transformations. Such transformations can be done on columns also.  Let us study the elementary row operations one by one.
1) In a matrix, a row or a column can be interchanged. For example,

a   b   c     can be interchanged as, d e f  or,  as,  a c b                                                                                                              d   e   f                                   

a   b   c             

d   f   e                                                                                                                                                                                 g   h   i

i    g   h            

g   i    h 

In the above example, the row interchange isdenoted as R2 <->R1 and the column interchange is denoted as C2 <-> C3.

2) A row or column can be modified by multiplying by a non- zero real number.For example,

a   b   c     can be modified as,   k[a]  k[b]   k[c]    or,  as,  k[a b c]                                                                                                             d   e   f                               d       e     f            k[d  e   f]                                                                                                                                                                                g   h   i                               g        h     i            k[g  h  I ]

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> kR1 and as C1 -> kC1

3) A row or column can be modified by multiplying by a non- zero real number. For example,

a   b   c     can be modified as,   a + kd    b + ke   c + kf      or,  as,   a + kb   b   c                                                                                                              d   e   f                                            d            e          f                        d + ke    e   f                                                                                                                                                                                 g   h   i                                            g             h          i                        g + kh    h   i

where, ‘k’ is a non-zero real number. These transformations are respectively denoted as R1 -> R1 + kR2 and as C1 -> C1 + kC2

These elementary transformations are extremely useful in further topics of matrices, like finding inverse of a matrix.For example, if A is an invertible matrix and B is its inverse, the formulas are,

A = I A and I = BA where I is the identity matrix of the same order.

Start with the equation A = IA.

Plug in the given matrix for A only on the left side and write only the identity matrix times A on the right. That is let the symbol A or the right remain as symbol A.

By repeated elementary row transformations, try to reach in the equation form I = BA. Then the matrix represented by B is the inverse of A.