Gaussian Elimination

Gaussian elimination is an algorithm to solve systems of equations.

Process to solve problems:

• Determine a “pivot,” i.e. the top leftmost nonzero number that isn’t in a row/column already used. If necessary, scale this whole row to make the pivot entry 1.

• For each row below the pivot row multiply the pivot row by the leading entry and subtract: Each row below the pivot row will be left with zeros in that column.

• Repeat until there are no more pivots. At this point, the matrix will have all 0s in the lower left triangle.

For example, starting with the above matrix, we have

Here, the pivots are shown in blue. Now let’s look at that last row--the equation it represents is z = 1. Then we can find and directly.

Gaussian elimination provides solutions to matrix equations of the form Ax = b where A is the matrix of

coefficients, x is the matrix of variables, b and is the matrix of RHSs (right hand side).