Row Operations
In order to develop an efficient algorithm for solving linear systems of equations, we need to determine operations on the systems and corresponding augmented matrices that do not change the solution sets. It turns out that there are three basic operations that are safe to perform:Q1 (Scaling): | Multiply both sides of an equation by a non-zero constant. | |
Q2 (Interchange): | Interchange two equations. | |
Q3 (Replacement): | Replace one row by the sum of of itself and a multiple of another row. |
These operations do not alter the set of solutions since the restrictions on the variables x_{1}, x_{2}, … , x_{n} given by the new equations imply the restrictions given by the old ones (that is, we can undo the manipulations made to retrieve the old system). Therefore, these three row operations are reversible.
At the same time, we observe that these operations really only involve the coefficients of the variables and the right-hand sides of the linear equations. So row operations can be applied to corresponding augmented matrices, but not only to linear systems. Actually, these operations are safe to apply for any matrix, not merely to augmented matrix. Hence, two matrices are called equivalent if there exists a sequence of elementary row operations that transfer one matrix into the other.