Practice problems for these concepts can be found at:

- Systems of Equations and Inequalities Solved Problems for Intermediate Algebra
- Systems of Equations and Inequalities Supplementary Problems for Intermediate Algebra

We previously employed the addition (elimination) method to solve systems of linear equations. In that process, various operations were performed to alter the coefficients of variables in equations. Since our attention was primarily on the coefficients of the variables, we can streamline the process by writing these coefficients only in an orderly array. The array we shall employ is called a matrix. A *matrix* is a rectangular array of elements (entries). The *elements* are usually numbers or letters. The elements or entries are displayed in rows and columns within brackets or parentheses. The following illustrate the symbolism normally used.

The *size, dimension*, or *order* of a matrix is specified by stating the number of rows followed by the number of columns. The size, dimension, or order of the above matrices is 3 × 3 (read 3 by 3), 2 × 3 (read 2 by 3), and 2 × 1 (read 2 by 1), respectively. A *square matrix* has the same number of rows and columns.

We now illustrate how matrices are used to solve a system of linear equations. Consider the system

The matrix is called the *coefficient matrix* of the system. It simply consists of the coefficients of the variables in the equations. The matrix is called the *constant matrix* of the system. It is composed of the constants in the right members of the equations. The matrix is called the *augmented matrix* of the system. It consists of the coefficient matrix of the system with the constant matrix of the system annexed on the right.

See solved problem 7.11.

Recall that equivalent equations have the same solution. *Equivalent systems of equations* likewise have the same solution(s). The associated augmented matrices of equivalent systems are *row-equivalent matrices*. Row equivalent matrices are obtained by the following operations.

**Elementary Row Operations**

- Interchange any two rows.
- Multiply each element of any row by a nonzero constant.
- Add a multiple of one row to another row.

We now illustrate the operations stated above and introduce notation that will help us represent those operations.

See solved problem 7.12.

The next concept needed is that of *row echelon form*. The following matrices are in row echelon form.

A matrix having the following characteristics is in row echelon form.

**Row Echelon Form**

- All rows that contain only zeros are positioned at the bottom of the matrix.
- If a row is not all zeros, the first nonzero element is a 1.
- Each leading 1 is at least one column to the right of the leading 1 of the preceding row.

Normally row echelon form is accomplished by first obtaining a 1 in row 1, column 1. Use elementary row operations to next transform the remaining entries in column 1 to zeros. Then obtain a 1 in row 2, column 2 and zeros in the rows below row 2 in column 2. Obtain a 1 in row 3, column 3 and zeros below row 3 in column 3. Continue until row echelon form is obtained. Roughly speaking the lower left portion of a matrix in row echelon form is all zeros, while entries on the main diagonal, which begins at row 1, column 1, and ends at row *n*, column *n*, consist of ones.

We now illustrate the techniques employed to transform a matrix to row echelon form.

See solved problem 7.13.

We now employ the concepts given above to solve systems of linear equations using matrices. The process is summarized below. The method is called the *Gaussian elimination method*.

**Method for Solving Linear Systems Using Matrices**

- Form the augmented matrix of the system.
- Use elementary row operations to transform the augmented matrix to row echelon form.
- Write the system of equations that corresponds to the echelon form matrix.
- Use back substitution to solve the system.
- Check the solution in the original equations.

See solved problem 7.14.

Practice problems for these concepts can be found at:

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