Gauss-Jordan Elimination Method

# What is it?

In Mathematics, the Gauss-Jordan elimination method is used in Linear Algebra as an row reduction algorithm for solving Linear Systems, and finding the inverse of a matrix. All operations are made on the matrix of coefficients, using the three Elementary Operations of Matrix.

The objective is to reduce the matrix to a reduced-row echelon form, also known as row canonical form, which will automatically solve the coefficients for each corresponding value. A matrix is in reduced-row echelon form when all conditions are satisfied:

• All rows with zero entries are at the bottom.

• The leading entry of a row is to the right of the leading entry of the row above it.

• The leading entry of any non-zero row is equal to $1$.

• All other entries in the column containing a leading entry are equal to $0$.

What is the leading entry?

When working with matrices, the leading entry refers to the first non-zero entry in a row. It’s also called the pivot entry.

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Examples ofform matrix

If the matrix satisfies all four conditions said above, it’s characterized as a reduced-row echelon form matrix. Some examples are:

$$A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 3\\ 0& 0& 0\end{array}\right)B=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

And some examples which are NOT satisfying the requirements. The matrix $C$ violates the second and third condition, and $D$ violates the fourth condition.

$$C=\left(\begin{array}{ccc}0& 7& 3\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)D=\left(\begin{array}{ccc}1& 7& 3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

Work with identity matrices

In general, if found easier, one could just transform the matrix to a identity matrix, which is a matrix with all elements of the main diagonal equal to $1$ and the rest filled with $0$.

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