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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