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We will give an algorithm, called row reduction or Gaussian elimination, which demonstrates that every matrix is row equivalent to at least one matrix in reduced row echelon form.

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What has happened geometrically is that the original blue line has been replaced with the new blue line y = 1. We can think of the blue line as rotating, or pivoting, around the solution ( 1,1 ) . We used the pivot position in the matrix in order to make the blue line pivot like this. This is one possible explanation for the terminology “pivot”.

We will solve systems of linear equations algebraically using the elimination method. In other words, we will combine the equations in various ways to try to eliminate as many variables as possible from each equation. There are three valid operations we can perform on our system of equations:

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The uniqueness statement is interesting—it means that, no matter how you row reduce, you always get the same matrix in reduced row echelon form.

When deciding if an augmented matrix is in (reduced) row echelon form, there is nothing special about the augmented column(s). Just ignore the vertical line.

If an augmented matrix is in reduced row echelon form, the corresponding linear system is viewed as solved. We will see below why this is the case, and we will show that any matrix can be put into reduced row echelon form using only row operations.

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Solving equations by elimination requires writing the variables x , y , z and the equals sign = over and over again, merely as placeholders: all that is changing in the equations is the coefficient numbers. We can make our life easier by extracting only the numbers, and putting them in a box:

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In the previous subsection we saw how to translate a system of linear equations into an augmented matrix. We want to find an algorithm for “solving” such an augmented matrix. First we must decide what it means for an augmented matrix to be “solved”.

We can visualize this system as a pair of lines in R 2 (red and blue, respectively, in the picture below) that intersect at the point ( 1,1 ) . If we subtract the first equation from the second, we obtain the equation 2 y = 2, or y = 1. This results in the system of equations:

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That is to say, the columns of the augmented matrix are exactly the column-vectors appearing in the equation. The column of constants in the equation, on the right of the “= ” sign, becomes the augmentation column of the matrix.

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This is called an augmented matrix. The word “augmented” refers to the vertical line, which we draw to remind ourselves where the equals sign belongs; a matrix is a grid of numbers without the vertical line.

An augmented matrix corresponds to an inconsistent system of equations if and only if the last column (i.e., the augmented column) is a pivot column.

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It will be very important to know where are the pivots of a matrix after row reducing; this is the reason for the following piece of terminology.

When we wrote our row operations above we used expressions like R 2 = R 2 − 2 × R 1 . Of course this does not mean that the second row is equal to the second row minus twice the first row. Instead it means that we are replacing the second row with the second row minus twice the first row. This kind of syntax is used frequently in computer programming when we want to change the value of a variable.

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