Now that we can write systems of equations in augmented matrix form, we will examine the various
row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to
row-echelon form , in which there are ones down the
main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
We use row operations corresponding to equation operations to obtain a new matrix that is
row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
In any nonzero row, the first nonzero number is a 1. It is called a
leading 1.
Any all-zero rows are placed at the bottom on the matrix.
Any leading 1 is below and to the right of a previous leading 1.
Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the
coefficient matrix to row-echelon form and do back-substitution to find the solution.
Multiply a row by a constant. (Notation:
$\text{\hspace{0.17em}}c{R}_{i}$ )
Add the product of a row multiplied by a constant to another row. (Notation:
$\text{\hspace{0.17em}}{R}_{i}+c{R}_{j})$
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.
Gaussian elimination
The
Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix
$\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ with the number 1 as the entry down the main diagonal and have all zeros below.