1.  SOLVE SYSTEMS OF EQUATIONS USING MATRICES. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 9.3 Matrices and Systems of Equations

2. Matrices Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The system can be expressed as where we have omitted the variables and replaced the equals signs with a vertical line.

3. Matrices Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley A rectangular array of numbers such as is called a matrix (plural, matrices). The matrix is an augmented matrix because it contains not only the coefficients but also the constant terms. The matrix is called the coefficient matrix.

4. Matrices continued Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The rows of a matrix are horizontal. The columns of a matrix are vertical. The matrix shown has 2 rows and 3 columns. A matrix with m rows and n columns is said to be of order m  n. When m = n the matrix is said to be square.

5. Gaussian Elimination with Matrices Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Row-Equivalent Operations 1. Interchange any two rows. 2. Multiply each entry in a row by the same nonzero constant. 3. Add a nonzero multiple of one row to another row. We can use the operations above on an augmented matrix to solve the system.

6. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve the following system: First, we write the augmented matrix, writing 0 for the missing y-term in the last equation.  Our goal is to find a reduced row-echelon form matrix. Each column contains a 1 on the diagonal and has 0’s everywhere else.

7. Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Which of the following matrices are in reduced row-echelon form? a)b) c)d)

8. Gauss-Jordan Elimination Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley We perform row-equivalent operations on a matrix to obtain a until we have a matrix in reduced row-echelon form. Example: Find the solution.

9. Special Systems Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley When a row consists entirely of 0’s, the equations are dependent. For example, in the matrix the system is equivalent to

10. Special Systems Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley When we obtain a row whose only nonzero entry occurs in the last column, we have an inconsistent system of equations. For example, in the matrix the last row corresponds to the false equation 0 = 9, so we know the original system has no solution.

11. Practice Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve the system of given systems of equations.