Math-2 (honors) Matrix Algebra Lesson 11.4

What you’ll learn Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

Vocabulary Matrix: A rectangular arrangement of numbers in rows and columns. Dimensions (order): Of a matrix with 3 rows and 2 columns is: 3 x 2 In general we say: m x n where: “m” = # of rows “n” = # of Columns

What is a Matrix? A rectangular arrangement of numbers in rows and columns 4 5 1 3 -2 5 What is the order or dimensions of this matrix?

Matrix Vocabulary Each element, or entry, aij, of the matrix uses double subscript notation. Row subscript: is the 1st letter (i) Column Subscript: is the 2nd letter (j) Example: Element aij is in the ith row and jth column. Is “shorthand” for a matrix with ‘i’ rows and ‘j’ columns. (‘i’ x ‘j’)

Vocabulary 5 -2 3 1 Elements: numbers in the matrix What is the “order” of this matrix ? 1,1 1,2 3 1 2,2 2,1 Equal matrices: have same “order” and each corresponding element is equal. Your turn: 1. What number is 2. What number is

Matrices can be HUGE ! 5 columns 4 rows Dimension: m rows x n columns 4 5 7 8 1 3 6 2 -2 5 -1 7 -3 3 -5 2 8 4 rows Dimension: m rows x n columns 4 x 5

Vocabulary Scalar: A real number (a constant) that is multiplied by every element in the matrix. Scalar Multiplication: The process of multiplying every element in the matrix by a scalar (constant). Let: ‘A’ represent the matrix Then: 3A = 3

Multiplying by a constant (also called a “Scalar Multiplication”) 3 4 9 1 7 -2 5(3) 5(4) 5(9) 5(1) 5(7) 5(-2) 5 = 15 20 45 5 35 -10 =

Your turn: 3. Write the result of the following: 2 –3 7 2 -4

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 ? + = A + B Matrices are added “corresponding element” to “corresponding element”

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 (2+3) + =

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 5 + =

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 5 (–3+1) + =

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 5 -2 + =

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 5 -2 + = (7-3)

Basic Operations: Addition 2 –3 7 2 3 1 -3 5 5 -2 + = 4

Basic Operations: Addition & Subtraction 2 –3 7 2 3 1 -3 5 5 -2 + = 4 (2+5)

Basic Operations: Addition & Subtraction 2 –3 7 2 3 1 -3 5 5 -2 + = 4 7

Basic Operations: Addition 2 –3 7 2 3 1 7 -3 5 6 ? + = CAN’T DO THIS!!!!!! (must be the same order for addition/subtraction)

Your turn: 4. Write the result of the following: 2 –3 2 7 2 4 3 1 7 -3 5 6 +

m x n (times) n x p = m x p m x p Order of the “Product” of Two Matrices (2 matrices multiplied by each other): Matrix A x Matrix B = AB m x n (times) n x p = m x p m x p Must be equal !!

Matrix Multiplication A x B = ? 2 –3 7 2 3 1 -3 5 ? x = 2 x 2 2 x 2 2 x 2 = equal m x p

What is the dimension of the product? 3 1 5 -3 5 4 2 –3 7 2 x ? = 2 x 2 2 x 3 2 x 3 = equal m x p

What is the dimension of the product? 3 1 -3 5 4 6 2 –3 7 2 x ? = 2 x 2 3 x 2 NOTequal !!!!!!! CAN’T DO THIS!!!!!!

So, how do you multiply matrices? 1 3 4 2 2 -1 1 -2 x = 1,1 1,2 2,1 2,2 Can you multiply? What is order of the answer matrix?

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = (1*2+3*1) 1,1 1,2 2,1 2,2 What is the “address” Of this element? 1,1: (1st row, 1st column)

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 1,1 1,2 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 (1*-1+3*-2) 1,1 1,2 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 -7 1,1 1,2 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 -7 1,1 1,2 (4*2+2*1) 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 -7 1,1 1,2 10 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 -7 1,1 1,2 (4*-1+2*-2) 10 2,1 2,2

So, how do you multiply matrices? 2 -1 1 -2 1 3 4 2 x = 5 -7 1,1 1,2 -8 10 2,1 2,2

Your turn: 5. Write the product of the two matrices. 3 1 5 -3 5 4 2 –3 7 2 x ? = 2 x 2 2 x 3 2 x 3 = equal m x p

Is Matrix Multiplication commutative? B = AB -7 10 -8 2 -1 1 -2 1 3 4 2 = x Your Turn: 1 3 4 2 2 -1 1 -2 x = ? 7. Write the product of the two matrices. 8. Does AB = BA ? B * A = AB

Matrix Multiplication using the Ti-84 2nd Matrix Edit, Enter order Enter elements 2nd matrix Edit (scroll to “B” matrix and enter elements Clear screen (2nd “quit”) 2nd matrix, scroll to desired matrix and enter Enter operation desired “*” 2nd matrix, scroll to other matrix and enter, then enter again.