Question

Prove that any matrix A can be transformed into a matrix B in row-reduced echelon form using elementary row operations.

Answer 1

We have elementary row operations ofone of the 3 types:1. permutation of 2 rows
2. multiplying the row by somescalar
3. adding to one row, another onemultiplied by some scalar.
Each of these operations are in oneto one correspondence with left matrix multiplying by elementary matrices of
next type.
1. identity matrix with permutedi-th and j-th row.
2. matrix diag{1,1,…1,a,1,…,1},where i-th diagonal element is some nonzero a.
3. identity matrix with oneadditional out of the diagonal element a on the (i,j) – th place, and i,j are
distinct.
When using Gauss method, we'llobtain a row echelon form in a result, and during it we use an elementary
operations of 3 mentioned types. As matrix A and B are equivalent by viewing
that B=E₁E₂…E_nA, and E_i areelementary matrices (they are invertible), and each of the matrices correspond
to one of the mentioned elementary row operations we conclude that matrix A can
be transformed into a matrix B in row-reduced echelon form using elementary row
operations.