Gauss-Jordan Elimination
Gauss-Jordan Elimination Name__________________
Accelerated Precalculus Period_____Date_________
An algorithm is a set of rules for solving a problem in a finite number of steps. Another algorithm for solving a system of equations is called Gauss-Jordan elimination. Although it is cumbersome for solving small systems, it works well for larger systems. The objective it to always eliminate x from all but the top equation. Then eliminate y from all but the second. Eliminate z from all but the third, etc. It looks like this:
Consider the system x ( 2y + 3z = 9 First write the system as a
(x + 3y = (4 coefficient matrix augmented
2x ( 5y + 5z = 17 with the constants:
So…[pic] Add the top two equations and we get
[pic] [pic]
[pic] [pic]
Once the first column is finished, use the second row as a “pivot row “ for the next column.
[pic] [pic]
[pic] Add the bottom two rows.
[pic] Divide the last row by 2 and we now have z.
[pic] Multiply the last row by (9 and add it to the top.
[pic] What happened here?
[pic] Do you see how the matrix tells us the solution?
Gauss-Jordan can also be used to find inverses. If we wanted to find the inverse of
[pic], we could set up the augmented matrix: [pic] .
Why does this work?
1. Solve these using Gauss Jordan elimination.
A) x + y + z = 8 B) x + 3y ( z = 11 C) x + y = (2
2x + y ( 3z = (5 2x + 2y ( 5z = 6 y + z = 2
2x ( 3y + 2z = 11 (x + y + 2z = 5 x + z = 6
2. Find these inverses:
A) [pic] B) [pic] C) [pic]
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