1 4 3 A Scalar Multiplication

Transcription

1 1 Matrices A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets. In a matrix, the numbers or data are organized so that each position in the matrix has a purpose. Each value in the matrix is called an element. A matrix is usually named using an uppercase letter A A matrix can be described by its dimensions. A matrix with m rows and n columns is an m x n matrix (read m by n ). Matrix A is a 2 X 3 matrix because it has 2 rows and 3 columns. a 12 refers to an element of A that is in row 1 column 2, a12 4. Operations with Matrices Matrices can be added or subtracted if they have the same dimensions. The corresponding elements are added or subtracted. Addition a b e f a e b f g h c g d h Subtraction a b e f a e b f g h c g d h You can multiply any matrix by a constant called a scalar. When you do this, you multiply each individual element by the value of the scalar. This operation is called scalar multiplication. Scalar Multiplication a b ka kb k kc kd For any matrices A, B, and C for which the matrix sum and product are defined and any scalar k, the following properties are true. Commutative Property of Addition A + B = B + A Associative Property of Addition (A + B) + C = A + (B + C) Scalar Distributive Property k(a + B) = ka + kb or (A + B)k =ka + kb And multi-step operations can be performed on matrices. The order of operations is the same as with real numbers.

2 2 The drink menu from a fast-food restaurant is shown below. The store owner has decided that all of the prices must be increased by 10%. Drink Small Medium Large Soda $0.95 $1.00 $1.05 Iced Tea $0.75 $0.80 $0.85 Lemonade $0.75 $0.80 $0.85 Coffee $1.00 $1.10 $1.20 a. Write matrix C to represent the current prices. State the dimensions of matrix C. b. Use scalar multiplication to determine matrix N to represent the new prices. State the dimensions of matrix N. c. What is P = N C? What does P represent in this situation? d. What is p 32 and what does it represent?

3 3 Matrix Multiplication You can multiply two matrices A and B if and only if the number of columns in A is equal to the number of rows in B. The product of two matrices is found by multiplying columns and rows, summing products as follows: A B = AB a b e f ae bg af bh g h ce dg cf dh Using the below matrices, A B C D 4 5 Find each product BA and AB Ex 2: CD and DC Ex 3: AC and CA *Is matrix multiplication commutative? Justify your answer.

4 4 Inverse Matrices and Systems of Equations Identity Matrix is a square matrix that, when multiplied by another matrix, equals that same matrix. 2 X 2 Identity Matrix 3 X 3 Identity Matrix I I Identity Matrix for Multiplication a b a b a b Two n x n matrices are inverses of each other if their product is the identity matrix. If matrix A has an inverse symbolized 1 A, then 1 1 A A A A I Determine whether the matrices are inverses. a. 4 2 A B b. 3 5 C D

5 5 a b The inverse of a matrix A is 1 1 d b A ad bc c a where the determinant ad bc 0. If the determinant of a matrix is 0, the matrix does not have an inverse. *Note: If the matrix is not square, the determinant and inverse cannot be found. Find the inverse of matrix P, if it exists. 7 5 P 2 1 Ex 2: Using your graphing calculator, find the inverse of matrix A, if it exists A Matrix Equations: Matrices can be used to represent and solve systems of equations. Consider the system of equations below. You can write a matrix equation to solve the system. x2y9 x 2y 9 3x6y 3 3x 6y 3 Write the left side of the matrix equation as the product of the coefficient matrix and the variable matrix. Write the right side as a constant matrix. A X = B 1 2 x y 3 Then solve the matrix equation using AX B A AX IX A B 1 1 X 1 A B 1 A B 1 A.

6 6 10x 7y 59 Solve the system using an inverse matrix. 6x 5y 63 Ex 2: 4x 5y 6z 14 Solve the system using an inverse matrix and a graphing calculator. 3x 2y 7z 47 7x 6y 8z 15 Ex 3: The sum of three numbers is 14. The largest is 4 times the smallest, while the sum of the smallest and twice the largest is 18. Find the numbers. Augmented Matrices and Systems of Equations An augmented matrix contains the coefficient matrix with an extra column containing the constant terms. You can use a graphing calculator to reduce the augmented matrix so that the solution of the system of equations can be easily determined. Step 1: Write the augmented matrix from the system of equations and enter it into the calculator. Step 2: Find the reduced row echelon form (rref) using the graphing calculator. *View the reduced row echelon form matrix as an augmented matrix, where the identity matrix is now the coefficient matrix and the last column of constants are the solutions to the system of equations. Write an augmented matrix for each system of equations. Then solve with a graphing calculator. 4y6x10 2x7y22

7 7 Ex 2: Write an augmented matrix for each system of equations. Then solve with a graphing calculator. 3x 5y 2z 7 4x 3y 5z 19 5x 4y 7z 15 Application: Matrices and Systems of Equations The admission fee for the Coastal Carolina Fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? Ex 2: A cashier has 25 coins consisting of nickels, dimes, and quarters with a value of $4.90. If the number of dimes is 1 less than twice the number of nickels, how many of each type of coin does she have? Ex 3: Sully, Krista, and Steve are combining their savings for a flight to Cancun. Together, they have $680. Sully has saved $20 less than Krista. Steve has managed to triple Krista and Sully s savings combined. How much has each contributed to the total savings?

8 8 Systems of Equations without a Unique Solution An Inconsistent System 3x2y4 6x 4y 7 A System with Infinitely Many Solutions 9x 3y 24 3x y8 2x3z 3 Solve the system using the graphing calculator. 4x 3y 7z 5 8x 9y 15z 9 3x 2y 4z 1 Ex 2: Solve the system using the graphing calculator. x y 2z 3 2x 3y 6z 8