Lesson 11-1 Matrix Basics and Augmented Matrices Objective: To learn to solve systems of linear equation using matrices.
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” EExample: is a 2 x 3 matrix
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
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)
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.
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.
Practice: 1. 4x + 6y = x - 4y + 2z = x - 5y + 5z = 10 8x - 2y = 7 2x - 2y + 6z = 10 5x - 5z = 5 2x + 2y + 2z = -2 5y + 10z = 0
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.
Adding Matrices To add matrices that are the same size, add the elements in each position.
Adding Matrices Example: = =
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.
Scalar Multiplication of Matrices Example: x 11 2 x = 2 x -9 2 x 6 = x -4 2 x scalar
Scalar Multiplication of Matrices Try: = 1/3 -9 1
Scalar Multiplication of Matrices Answer: = /
Adding Matrices Try: =
Adding Matrices Answer: = =