Row reduction is the process of performing row operations to transform any matrix into (reduced) row echelon form. In reduced row echelon form, each successive row of the matrix has less dependencies than the previous, so solving systems of equations is a much easier task. The idea behind row reduction is to convert the matrix into an "equivalent" version in order to simplify certain matrix computations. Its two main purposes are to solve system of linear equations and calculate the inverse of a matrix.
Carl Friedrich Gauss championed the use of row reduction, to the extent that it is commonly called Gaussian elimination. It was further popularized by Wilhelm Jordan, who attached his name to the process by which row reduction is used to compute matrix inverses, Gauss-Jordan elimination.
A system of equations can be represented in a couple of different matrix forms. One way is to realize the system as the matrix multiplication of the coefficients in the system and the column vector of its variables. The square matrix is called the coefficient matrix because it consists of the coefficients of the variables in the system of equations:
An alternate representation called an augmented matrix is created by stitching the columns of matrices together and divided by a vertical bar. The coefficient matrix is placed on the left of this vertical bar, while the constants on the right-hand side of each equation are placed on the right of the vertical bar:
The matrices that represent these systems can be manipulated in such a way as to provide easy-to-read solutions. This manipulation is called row reduction. Row reduction techniques transform the matrix into reduced row echelon form without changing the solutions to the system.
The reduced row echelon form of a matrix denoted is a matrix of equal dimensions that satisfies the following:
- The leftmost non-zero element in each row is . This element is known as a pivot.
- Any column can have at most pivot. If a column has a pivot, then the rest of the elements in the column will be .
- For any two columns and that have pivots in rows and respectively, if pivot in is to the left of pivot in , then is above . In other words, for any two pivots and , if is to the right of , then is below .
- Rows that consist of only zeroes are in the bottom of the matrix.
To convert any matrix to its reduced row echelon form, Gauss-Jordan elimination is performed. There are three elementary row operations used to achieve reduced row echelon form:
- Switch two rows.
- Multiply a row by any non-zero constant.
- Add a scalar multiple of one row to any other row.
Find using Gauss-Jordan elimination, where
The leftmost element in the first row must be 1, so the first row is divided by 2:
The top left element is a pivot, so the rest of the elements in the first column must be 0. This can be done by subtracting the first row from the second row. Furthermore, the first row can be added to the third row to obtain the necessary 0s in the first column:
Now that the leftmost column is the middle element can be made 1 by dividing the second row by 3:
The top and bottom elements in the second column can be made 0 with the appropriate row operations:
With the middle column now the method proceeds to the third column by dividing the third row by 15:
In the final step of the process, multiples of the third row are added to the first and second rows so that the last column becomes
Given an equation with a unique solution for , it is possible to solve the equation by calculating and then evaluating However, it takes much more time to calculate and then multiply by than just to solve for . For that reason, is only useful to compute when it is needed in itself, perhaps to solve general equations.
Perform row reduction on the matrix to obtain something of the form . Then, provides a solution to the inverse problem.