Prove that any elementary row (column) operation of type 1 can beobtained by a s

Question

Prove that any elementary row (column) operation of type 1 can beobtained by a succession of three elementary row (column)operations of type 3 followed by one elementary row (column)operation of type 2.

NOTES:
type 1: interchanging any two rows (columns) of A;
type 2: multiplying any row (column) of A by a nonzero scalar;
type 3: adding any scalar multiple of a row (column) of A toanother row (column).

Explanation / Answer

   Suppose we want to  interchange the ith row with thejth row :    Now we perform the following operations of type 2 andtype 3 to achieve the above :    First operation :   add theith row to the jth row , it is oftype 3 ,                             Rj   -> Rj +Ri Second operation :   now subtract the jth row from the ith row , it is oftype 3 ,                             Ri -> Ri - ( Rj +Ri ) = - Rj Third operation :   now add the ith row to the jth row , it is oftype 3 ,                             Rj ->   Rj + Ri - Rj = Ri Fourth operation : Now multiply theith row with (-1) , it is of type 2 :                                Ri -> (-1)* ( - Rj ) = Rj   hence, now the jth row is Ri and the ith row is Rj

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