Row Operations

Consider the linear system

We can see directly that the solution is .

We can represent it in matrix form as

or, in short,

where is the matrix of coefficients, is the right hand side vector and is the vector of unknowns.

If we multiply one equation by a constant (for example, multiply first equation by ) we obtain an equivalent system, i.e. a system with the same solution. In our example, we can see that

has the solution .

If we add two equations together, and replace the second equation by the result, we obtain an equivalent system also. For example, first equation plus the second give

and, when replacing the second equation by this result, we get

This system has the same solution .

In conclusion, multiplying one equation by a constant or replacing one equation by the sum of itself plus another equation lead to equivalent systems.

If we combine these two operations into a single step we conclude that we can replace one equation by the sum of itself plus a multiple of another equation without modifying the solution of the system. For example, multiplying the first equation of the system () by and adding it to the second equation leads to the equivalent system