1. Lesson 11-1 Matrix Basics and Augmented Matrices Objective: To learn to solve systems of linear equation using matrices.

2. Matrices  A rectangular array of numbers is called a matrix (plural is matrices)  I It is defined by the number of rows (m) and the number of columns (n) “m by n matrix” EExample: is a 2 x 3 matrix 1 0 5 2 3 4

3. Matrices  Each number in the matrix has a position A =  Each item in the matrix is called an element a 11 a 12 a 13 a 21 a 22 a 23

4. What is the dimension of each matrix? 3 x 3 3 x 5 2 x 2 4 x 1 1 x 4 (or square matrix) (Also called a column matrix) (or square matrix) (Also called a row matrix)

5. Augmented Matrices  System of Linear Equation  expressed in a matrix : Augmented matrix has the coefficients of all the variables (in order) along with the answers in the last column.

6. Using the Calculator to Solve  [2 nd ] [matrix] EDIT[ENTER]  MATRIX [A] IS A 3 x 4 matrix (3 rows x 4 columns)  then enter all the data into the matrix  Once data is entered, quit then  [2 nd ] [matrix] MATH  scroll down to B: rref [ENTER] [2 ND ] [MATRIX] [A] [ENTER]  You will get a new matrix - the last column is your answer for x, y and z.

7. Practice:  1. 4x + 6y = 0 2. 6x - 4y + 2z = -4 3. 5x - 5y + 5z = 10  8x - 2y = 7 2x - 2y + 6z = 10 5x - 5z = 5  2x + 2y + 2z = -2 5y + 10z = 0

8. Adding Matrices  In order to add matrices each one must have the same number of rows and also the same number of columns. (You can add a 3 x 2 matrix to another 3 x 2 matrix, but not to a 1 x 5 matrix).  Matrix equality occurs when 2 matrices have the same dimensions and the same entries.

9. Adding Matrices  To add matrices that are the same size, add the elements in each position.

10. Adding Matrices  Example: 2 1 1 5 2+1 1+5 3 6 0 -1 + 2 0 = 0+2 -1+0 = 2 -1 3 4 -1 1 3-1 4+1 2 5

11. Scalar Multiplication of Matrices  The first type of multiplication we will investigate is called scalar multiplication.  In scalar multiplication each element in a matrix is multiplied by a number, called a scalar.

12. Scalar Multiplication of Matrices Example: 11 -5 2 x 11 2 x -5 22 -10 2 -9 6 = 2 x -9 2 x 6 = -18 12 -4 3 2 x -4 2 x 3 -8 6 scalar

13. Scalar Multiplication of Matrices  Try: -4 -5.3 2 3 3.1 0 6 = 1/3 -9 1

14. Scalar Multiplication of Matrices  Answer: -4 -5.3 2 -12 -15.9 6 3 3.1 0 6 = 9.3 0 18 1/3 -9 1 1 -27 3

15. Adding Matrices  Try: -5 2 10 3 7 -1 + -2 4 = 8 9 -3 0

16. Adding Matrices  Answer: -5 2 10 3 -5+10 2+3 5 5 7 -1 + -2 4 = 7-2 -1+4 = 5 3 8 9 -3 0 8-3 9+0 5 9