2.4 - Setting Up and Solving Systems of Linear Equations

Augmented Matrix

To express a system of linear equations in matrix form, we extract the coefficients of the variables and the constants from the equations, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equals signs.

Elementary Row Operations

Can routinely be performed on a matrix representing a system of linear equations:

I. Interchange (reorder) two rows.
II. Multiply one row by a nonzero constant, 𝑘.
III. Add a multiple of one row to a different row.

Reduced row-echelon form

A matrix is in this form if it satisfies the following four conditions:

1. All zero rows (rows consisting entirely of zeros) are below all nonzero rows, if they exist.
2. The first nonzero entry from the left in each nonzero row must be a 1, called the leading 𝟏 for that row.
3. Each leading 1 is to the right of all leading 1's in the rows above it.
4. Each leading 1 is the only nonzero entry in its column.

Gauss-Jordan Elimination Method

1. Write the corresponding augmented matrix for the given system of linear
equations.
2. Interchange rows, if necessary, to obtain a nonzero number in the first row, first column.
3. Use a row operation to make the entry in the first row, first column a ' 1 '.
4. Use row operation(s) to make all other entries in the first column ' 0 '.
5. Interchange rows, if necessary, to obtain a nonzero number in the second row, second column. Use a row operation to make this entry 1. Use row operations to make all other entries in the second column zero.
6. Repeat Step 5 for Row 3, Column 3 and the third column. Continue moving along the main diagonal from upper left to lower right until you reach the last row, or until the number is zero, which cannot be removed, and the solution can easily be read.
At the completion of this process, the final matrix will be in reduced row-echelon form.

Solving Systems using Gauss-Jordan Elimination with Technology

The Gauss-Jordan Elimination Method can be performed using the TI-84 calculator, when the number of rows in your matrix is less than or equal to the number of columns. If you are not required to show each row operation, you can use this method to solve any system of linear equations.