Question #24291

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

Expert's answer

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=E_{1}E_{2}…E_{n}A, 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.

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=E

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.

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