Show that if matrices A and B have the same RREF, then A can be turned into B by a finite sequence of elementary row operations. Hint: Show that any elementary row operation can be undone (inverted,).

The answer is below

Step-by-step explanation:

A matrix is in row echelon form if it satisfies the following three conditions:

(i). All non-zero rows are above any zero rows.

(ii). Each leading entry in a row is in a column to the right of the leading entry of the row above it.

(iii). All the entries in a column below a leading entry are zeros.

The following operations on a matrix are called elementary row-operations:

1. Interchanging the ith and the jth rows, expressed as ri <--> rj.

2. Replacing the ith row by a multiple of itself expressed as ri --> krj, where k is a non-zero scalar.

3. Replacing the jth row by the sum of itself and a non-zero multiple of the ith row, expressed as rj --> rj + kr.

It may be observed that all the 3 elementary row-operations are reversible. Thus. ri <--> rj can be reversed by ri <--> rj,  ri --> kri can be reversed by ri -->(l/k)*ri, (here ri indicates the new ith row) and  rj --> rj + kr can be reversed by rj --> rj - krj.

Now, since both the matrices A and B are reduced to the same RREF C (say), hence reversing the elementary operations used to reduce B to C, we get B from C. Thus, A can first be converted to C and then to B.

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