4.1 Matrix Operations
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1 MATRICES
2 4.1 Matrix Operations Always read a matrix ROW by COLUMN # Rows: Dimension: # Columns: Numbers in the matrix are called entries. What is the entry in the 2 nd row and 3 rd column for the matrix above? Different Types of Matrices Name Example Dimensions Row Matrix x 4 Column Matrix x 1 Square Matrix x 3
3 Matrix Addition and Subtraction: Matrices must have the SAME dimensions. Add or subtract the corresponding entries. Example 1: Dimension of each matrix: Dimension of the answer matrix: Example 2: = Dimension of each matrix: Dimension of the answer matrix: Scalar Multiplication: Multiply the constant OUTSIDE the matrix to EACH entry inside the matrix. Example 3: = Dimension of the answer matrix: Scalar Multiplication combined with Addition or Subtraction: Example 4: = Dimension of each matrix: Dimension of the answer matrix: Solve the following matrix for x and y Corresponding entries are equal Example 5: 2 3x y =
4 4.2 Matrix Multiplication Matrix Multiplication: The number of columns in the first matrix must match the number of rows in the second matrix. If [A] has dimensions m x n If [B] has dimensions n x p The product of [A]x[B] will have dimensions m x p A: 2 X 3 B: 3 X 4 A: 3 X 2 B: 3 X 4 Dimension of [A]x[B]: Dimension of [A]x[B]: Example 6: Find AB A = B = Dim of [A]: Dim of [B]: Product Dim: Example 7: Find BA A = B = Dim of [A]: Dim of [B]: Product Dim: Example 8: Find AB + BC A = , B = , and C =
5 Use your calculator to add, subtract, multiply with matrices. To enter a Matrix in your calculator: 2 nd MATRIX EDIT ENTER (enter the dimensions of the matrix and the entries) To call up a Matrix in your calculator from the home screen: 2 nd MATRIX (highlight the matrix) ENTER A = , B = , and C = ) B (A + C) 2.) BA + BC
6 Application of Matrices: A health club offers three different membership plans. With Plan X, you can use all club facilities: the pool, fitness center, and racket club. With Plan Y, you can use the pool and fitness center. With Plan Z, you can only use the racket club facilities. The matrices below show the annual cost for a Single and a Family membership for the years 2012 through [A] [B] [C] single family single family single family plan X plan Y plan Z plan X plan Y plan Z plan X plan Y plan Z ) Determine a matrix that gives the price increase from 2012 to 2014 for each of the plans. 2) Determine a matrix that gives the total cost for all three years for each of the plans. 3) The health club offered a 3- year membership based on the 2012 rates. How much money does the 3- year membership save for each plan compared to paying the regular membership rate for each of the 3 years?
7 4.3 Determinants Determinant of a 2 x 2 matrix: a b a b det = c d = ad bc c d Determinant of a 3 x 3 matrix: det a b c d e f = g h i a b c d e f g h i = (aei + bfg + cdh) ( ceg + afh + bdi)
8 Solving a System of Equations with Matrices Use a matrix and a graphing calculator to solve the linear system: 2 nd MATRIX EDIT ENTER (edit matrix) 2 nd MATRIX MATH B rref( 2 nd MATRIX (select the matrix that you edited) 1) 2x y + 4z = 48 x + 2y + 2z = 6 x 3y + 4z = 54 Use the matrix: Solution matrix: x y z 2) x + y 2z = 9 2x + y + z = 0 x 2y + 6z = 21
9 4.3 Worksheet You may use your calculator for these problems. Evaluate the determinant = = = Use Matrices to solve the system of equations x + y z = 3 2x 3y + 4z = 23 3x + y 2z = 15 3x + 3y + 4z = 1 3x + 5y + 9z = 2 5x + 9y +17z = 4 5x + 3y 2z = 4 2x + 2y + 2z = 0 3x + 2y +1z = 1 2x 4y + 5z = 33 4x y = 5 2x + 2y 3z = 19 Applications: 1. Claire and Dale shopped at the same store. Claire bought 5 kg of apples and 2 kg of bananas and paid altogether $22. Dale bought 4 kg of apples and 6 kg of bananas and paid altogether $33. Use matrices to find the cost of 1 kg of bananas.
10 2. Ann and Billy both entered a quiz. The quiz had twenty questions and points were allocated as follows: P points were added for each correctly answered question Q points were deducted for each incorrect (or unanswered) question Ann got 15 questions correct and scored 65 points. Billy got 11 questions correct and scored 37 points. Use matrices to find the value of Q. 3. A community relief fund receives a large donation of $2800. The foundation agrees to spend the money on $20 school bags, $25 sweaters, and $5 notebooks. They want to buy 200 items and send them to schools in earthquake- hit areas. They must order as many notebooks as school bags and sweaters combined. How many of each item should they order? 4. An ultimate Frisbee team has to order jerseys, shorts, and hats. They have a budget of $1350 to spend on $50 jerseys, $20 shorts, and $15 hats. They want to buy 40 items in preparation for the oncoming season and must order as many jerseys as shorts and hats combined. How many of each item should they order?
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