Question

Suppose matrix Ais transformed into matrix Bby means of an elementary row operation. Is there an elementary row operation that transforms Binto A? Explain.

Short answer

For transforming the matrix B into A we can do the inverse of operations that we have used for transforming the matrix A into B.

Step-by-step solution

Row operation

In matrix there are three type of row operation for transforming the matrix:

1. Swapping the rows: In this operation we can swap any row with any other row.
2. Constant multiplications with row: In this operation we can multiply any non-zero scalar with any row.
3. Subtracting or adding the row: In this operation we can subtract or add another row.

Transforming A into B matrix

Suppose we have two matrixes A and B. For transforming the matrix A to B we can use three operations.

We can do swapping the rows like$R_i\leftrightarrow R_j$.

We can divide the row by scalar k like $R_i\to \frac{R_i}{k}$.

We can add the row like $R_i\to R_i-k{R}_j$.

Transforming B into A matrix

For transforming the matrix B into A we can do the inverse of operation that we have used for transforming the matrix A into B.

If we have used swapping the rows like$R_i\leftrightarrow R_j$ we can do the inverse like $R_j\leftrightarrow R_i$.

If we have used divide the row by scalar k like$R_i\to \frac{R_i}{k}$ we can do the inverse like.

$R_j\to \frac{R_i}{k}$

If we have used adding the row like$R_i\to R_i-k{R}_j$ we can do the inverse like $R_j\to R_j-k{R}_i$.

Hence, for transforming the matrix B into A we can do the inverse of operations that we have used for transforming the matrix A into B.