Gauss-Jordan Elimination

High School

Definition

It is also known as Gaussian elimination. A way to solve linear systems of equations. It takes an equation's augmented matrix and turns it into a reduced row echelon form through row operations. The reduced row echelon form has zeroes on its lower diagonal and the first nonzero number in each of the rows is 1.

Worked examples

\(\left[\begin{array}{cc|c}2 & 4 & 10 \)
\( 1 & -1 & 0\end{array}\right] \)→\( \left[\begin{array}{cc|c}1 & 0 & 1 \)
\( 0 & 1 & 2\end{array}\right]\)

Row operations transform the augmented matrix into reduced row echelon form, revealing \(x = 1\) and \(y = 2\).

\(\left[\begin{array}{ccc|c}1 & 2 & 1 & 5 \)
\( 0 & 1 & 3 & 7 \)
\( 0 & 0 & 1 & 2\end{array}\right] \)→\( \left[\begin{array}{ccc|c}1 & 0 & 0 & -3 \)
\( 0 & 1 & 0 & 1 \)
\( 0 & 0 & 1 & 2\end{array}\right]\)

Back-substitution with row operations clears above and below each leading 1 to get the solution.

Common mistakes

Stop at row echelon form (staircase with leading 1s) → Continue to reduced row echelon form (clear above each pivot) Gauss-Jordan requires zeros above and below each leading 1, not just below.
\(\left[\begin{array}{cc|c}2 & 4 & 10 \)
\( 1 & -1 & 0\end{array}\right] \)→\( \left[\begin{array}{cc|c}2 & 4 & 10 \)
\( 0 & -3 & -5\end{array}\right]\) (stop here) → Scale rows so each pivot is 1: \(\left[\begin{array}{cc|c}1 & 2 & 5 \)
\( 0 & 1 & \frac{5}{3}\end{array}\right]\) The first nonzero in each row must be 1 in reduced row echelon form.
Perform operations on the coefficient part only, ignoring the augmented column → Apply every row operation to the entire row, including the augmented column The augmented column holds the constants; omitting it gives wrong solutions.

Where you'll use it next

You'll use Gauss-Jordan elimination in linear algebra to solve systems, find matrix inverses, and determine rank. It extends to vector spaces, eigenvalue problems, and applied fields like engineering and computer graphics.

Found in 1 StudyPug lesson

Mastering Gaussian Elimination for Linear Systems

Grade 12 Math

Unlock the power of Gaussian elimination to solve complex linear systems efficiently. Learn step-by-step techniques, from creating augmented matrices to performing row operations and back-substitution.

12th Grade12th