In the following example, suppose that each of the matrices was the result of carrying an augmented matrix to reduced row-echelon form by means of a sequence of row operations.
That is, the resulting system has the same solution set as the original system. In particular, we bring the augmented matrix to Row-Echelon Form: Row-Echelon Form A matrix is said to be in row-echelon form if All rows consisting entirely of zeros are at the bottom.
If, in addition, each leading 1 is the only non-zero entry in its column, then the matrix is in reduced row-echelon form. Subtract multiples of that row from the rows below it to make each entry below the leading 1 zero. Each row then represents the coefficients of each variable in the order that they appear in the system of equations.
However, these methods work for systems of all sizes, so you have to choose which method is appropriate for which problem.
Take a look at the following system: To express this system in matrix form, you follow three simple steps: Write all the coefficients in one matrix first.
You can write any system of equations as a matrix. The setup appears as follows: Notice that the coefficients in the matrix go in order — you see a column for x, y, and z. For instance, in Step 2 you often have a choice of rows to move to the top.
Our strategy in solving linear systems, therefore, is to take an augmented matrix for a system and carry it by means of elementary row operations to an equivalent augmented matrix from which the solutions of the system are easily obtained.