In one line
Gauss–Jordan continues the elimination above the pivots as well, reducing the coefficient matrix to the identity so that the solution can be read directly from the last column, with no back-substitution.
Explained simply
Gauss elimination leaves a staircase and then you climb down solving one unknown at a time. Gauss–Jordan keeps cleaning until each equation has only one unknown with coefficient 1: \(x=\dots\), \(y=\dots\), \(z=\dots\). It is like tidying a room completely instead of half-way: more work at first, but nothing left to do at the end.
Technically, row operations reduce \([A|B]\) to \([I|X]\). Every pivot is made 1, and all other entries in its column (above and below) are made 0. The same procedure applied to \([A|I]\) gives \([I|A^{-1}]\), which is a practical way to compute an inverse. This is a self-learning topic.
How it works
- Write the augmented matrix.
- Make the first pivot 1 and clear the rest of column 1.
- Make the second pivot 1 and clear the rest of column 2 (both above and below).
- Repeat for column 3.
- Read the solution from the last column.
Key formula
\([A\,|\,B]\xrightarrow{\text{row operations}}[I\,|\,X]\); \(\ [A\,|\,I]\to[I\,|\,A^{-1}]\).
Quick example
Continue the Gauss elimination example \(x+y+z=6\), \(2x+3y+4z=20\), \(3x+2y+5z=22\), which reached \(\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&8\\0&0&4&12\end{array}\right]\).
\(R_3\to\frac14R_3\): last row \([0\ 0\ 1\,|\,3]\).
\(R_1\to R_1-R_2\): \([1\ 0\ -1\,|\,-2]\).
\(R_1\to R_1+R_3\): \([1\ 0\ 0\,|\,1]\); \(\ R_2\to R_2-2R_3\): \([0\ 1\ 0\,|\,2]\).
Result \(\left[\begin{array}{ccc|c}1&0&0&1\\0&1&0&2\\0&0&1&3\end{array}\right]\): \(x=1,\ y=2,\ z=3\).
Common mistakes
- Clearing only below the pivots (that is Gauss elimination).
- Forgetting to scale pivots to 1, so the last column is not the answer.
- Using column operations; only row operations are valid for solving equations.
How MU asks it
As a self-learning topic it may appear as a 5-mark part: "Solve by Gauss–Jordan method" or "Find \(A^{-1}\) by Gauss–Jordan method". Show every row operation, the final identity form, and read off the solution. Mentioning the difference from Gauss elimination in one line helps in theory parts.