Finite Math B Chapter 2 + Supplements: MATRICES 1 A: Matrices, Basic Matrix Operations (Lessons 2.3 & 2.4 pg 86 107) A matrix is a rectangular array of numbers like: Chapter 2 and Supplements MATRICES This matrix has rows and columns. When we want to describe a particular entry in an m-by-n matrix, we use the notation a i, j where ( i, j) ( row, column) position. For example, in the matrix above: 3 5 1 2 4 7 a 1,2 = a 2,3 = The number -1 is in the position. A matrix with an equal number of rows and columns is called a Square Matrix A matrix with only one row is called a Row Matrix. A matrix with only one column is called a Column Matrix. Example 1: State the size of each matrix. State if the matrix is a square, row, or column matrix when applicable. a) 1 3 4 8 b) 2 4 6 3 5 9 8 1 0 c) 9 3 2 8 1 8 d) 2 1 0 Adding/Subtracting Matrices If two matrices are the same size (mxn) then you can add or subtract them by combining corresponding elements.
Finite Math B Chapter 2 + Supplements: MATRICES 2 Example 2: Add or subtract the following matrices. Write undefined for expressions that are undefined. a. 2 4 3 0 5 9 4 8 b. 3 4 2 9 5 6 4 8 7 8 2 3 9 10 1 11 c. 3 4 1 5 1 0 7 8 1 7 3 6 4 d. 1 2 0 80 7 5 9 The Additive Inverse of a matrix X is a matrix X such that X + -X = 0 Example 3: What is the Additive inverse of the following matrices? a) 1 3 4 8 b) 9 3 2 8 1 8 Setting up a matrix to represent a real life problem. Example 4: A trendy garment company receives orders from three clothing shops. The first shop orders 25 jackets, 75 shirts, and 75 pairs of pants. The second shop orders 30 jackets, 50 shirts, and 50 pairs of pants.. The third shop orders 20 jackets, 40 shirts, and 35 pairs of pants. Display this information as a matrix where the rows represent shops and the columns represent types of garments.
Finite Math B Chapter 2 + Supplements: MATRICES 3 Example 5: You are planning dinner and decide to call several pizza shops and get prices. Gina s Pizzeria charges $12.16 for a large one-topping pizza, $1.15 for a 2-liter of soda, and $4.05 for a family sized salad. Toni s Pizza charges $10.86 for a pizza, $0.89 for soda, and $3.89 for salad. Display this information as a 3 x 2 matrix. You are also told that at Gina s an extra pizza topping is $0.50, but at Toni s it is only $0.30. In addition, at Gina s salad dressing is included, but at Toni s salad dressing is $1.00 extra. Write a new 3x2 matrix that shows the costs of a pizza with an additional topping and a salad with dressing. Example 6: The National League batting leaders for 2003 had the following batting statistics: The following show the statistics for the same three players at the end of the 2004 season. Find and label a matrix that displays the CHANGES in these statistics from 2003 to 2004. Notice that several of the statistics decreased. How will you show that in your matrix?
Finite Math B Chapter 2 + Supplements: MATRICES 4 B: Scalar Factors, Determinants, Graphing Calculators (Lessons 2.4 p96 and supplements) Multiplication by a Scalar Factor: Multiply each entry in the matrix by the factor b c xb xc x d e xd xe Example 1: Simplify. a. 0 3 8 5 b. 1 7 3 2 4 5 3 1 0 8 Example 2: Example 3:
Finite Math B Chapter 2 + Supplements: MATRICES 5 Determinants A determinant is a special number associated with a square matrix. The determinant has several useful applications in algebra and other advanced mathematics courses. Notation: Matrix Notation (square brackets) 2x2 a c b d 3x3 a b c d e f g h i Determinant Notation (straight brackets) 2x2 a c b d 3x3 a b c d e f g h i (This notation represents find the determinant of the matrix ) Also: det(a) or deta represents the determinant of matrix A 2x2 Matrices The determinant of a 2 x 2 matrix is found as follows: Example 4: Find the determinant of each 2 x 2 matrix a) 3 4 5 6 b) 1 3 6 7 Example 5: Evaluate. a) 2 3 5 10 b) 2 7 3 5
Finite Math B Chapter 2 + Supplements: MATRICES 6 3x3 Matrices The determinant of a 3x3 matrix is found as follows: This is easier to find using a little trick called The Rule of Sarrus : 1. Copy the first two columns next to column 3. 2. Multiply the three southeast (downward)diagonals. Find the sum. 3. Multiply the three northeast (upward) diagonals. Find the sum. 4. Determinant = (SE sum) (NE sum) bottom top Example 6: Find the determinant of each matrix. a) 1 3 5 2 1 3 4 5 2 b) 2 3 1 4 2 3 2 2 1 Example 7: Evaluate 1 2 3 1 2 3 3 2 1
Finite Math B Chapter 2 + Supplements: MATRICES 7 Using Technology Working with Matrices in the TI 83/TI83plus/TI84 Graphing Calculator **NOTE: If you have your OWN graphing calculator, you are permitted to use it on exams. Otherwise, you will NOT BE PROVIDED with a graphing calculator on exam day. There are no items on the exam that will require the use of this technology** TI-83 Matrix Menu MATRX 2 nd x -1 About the Matrix (MATRX) menu: NAMES used to paste the name of a matrix into the home screen or into a program. MATH contains all the operations that can be done with a matrix. EDIT where you set the size of the matrix and enter the elements. Note: Some TI 83 models do not require you to shift to access the matrix menu. BASIC OPERATIONS Use the graphing calculator to perform the following operations: 1. 2 4 3 0 5 9 4 8 2. 2 4 3 0 5 9 4 8 3. 2 4 1 5 9 4 4. 2 4 5 5 9
Finite Math B Chapter 2 + Supplements: MATRICES 8 DETERMINANTS In the matrix menu, use the EDIT screen to enter your matrix. Use 2 nd (MODE) to quit to the main screen. In the matrix menu, use the MATH screen to paste det( into your home screen and the NAMES screen to paste the name of the matrix. Close ) and hit enter Use the graphing calculator to find the determinant of each of the following matrices. 5. 2 3 5 10 6. 1 3 5 2 1 3 4 5 2 C: Multiplying Matrices (Lesson 2.4 pg 97 107) Let A be an m x n matrix Let B be an n x k matrix The product matrix AB is an m x k matrix. Example 1: The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist. a) A is 4 x 4, B is 4 x 2 b) A is 3 x 1, B is 3 x 1 c) A is 2 x 3, B is 1 x 2
Finite Math B Chapter 2 + Supplements: MATRICES 9 Multiplying Matrices Let A be an m x n matrix Let B be an n x k matrix To find the product: Multiply each element in the row of A by the corresponding element in the column of B, then add these products. The product matrix AB is an m x k matrix. SAMPLE 1: Find the product AB 2 3 1 A 4 2 2 1 B 8 6 A is 2 x 3 B is 3 x 1 -------> AB will be 2 x 1 Step 1: Multiply the elements of the first row of A and the corresponding elements in the column of B. 2(1) + 3(8) + -1(6) = 20 Step 2: Multiply the elements of the second row of A and the corresponding elements in the column of B. 4(1) + 2(8) + 2(6) = 32 Step 3: Write your solution as a 2 x 1 matrix. 20 32 SAMPLE 2: Find the product
Finite Math B Chapter 2 + Supplements: MATRICES 10 Example 2: Find the product. Write undefined for expressions that are undefined. a) 2 2 4 6 3 1 3 0 4 b) 2 1 3 5 8 2 c) 2 1 7 3 3 6 1 4 d) 0 2 2 2 1 1 4 3 0 1 0 2
Finite Math B Chapter 2 + Supplements: MATRICES 11 e) 1 1 2 2 3 3 2 1 3 2 4 5 2 4 5 f) 2 5 3 1 2 3 7 4 6 3 2 1 3 1 g) 2 2 4 6 1
Finite Math B Chapter 2 + Supplements: MATRICES 12 D: Geometric Transformations using Matrices We can use matrices to represent geometric figures: A transformation is a change made to a figure. The original figure is called the preimage, while the transformed figure is called the image. TRANSLATION When we slide a figure without changing the size or shape of the figure, it is said to be a translation. By using matrix addition, we can translate the vertices of a figure. y Example 1: Given a triangle ABC where A(-4,1), B (-2,5), and C(0,2), translate the preimage 5 units right and 3 units down. Then sketch the image Preimage Matrix + Translation Matrix = new image Example 2: Given a quadrilateral ABC where A(3,0), B(5,2), C(5,4), and D(2,4). Translate the preimage six units left and 2 units up.
Finite Math B Chapter 2 + Supplements: MATRICES 13 y DILATION A dilation is a transformation that changes the size of the preimage. Dilation factor x preimage matrix = new image Example3: Given a triangle ABC where A(-2,0), B(0,4) and C(2,1), increase the triangle by a factor of 2. Then sketch the image. Example 4: Given the quadrilateral with vertices A(4, 3), B(0,5), C(-10,14) and D(1, 1.5), increase the figure by 150%. State the vertices of the new image. REFLECTIONS (FLIPS) A reflection, or flip, is a transformation that creates symmetry on the coordinate plane. A reflection maps a point or figure in the coordinate plane to its mirror image using a specific line as its line of reflection. Usually we use the following lines of reflection: Reflection Matrix x Preimage Matrix = New Image Example 5: Given quadrilateral A(2,1), B(8,1), C(8,4), and D(5,5), find the coordinates of the image after a reflection: a) Across the y-axis b) Across y = x
Finite Math B Chapter 2 + Supplements: MATRICES 14 ROTATIONS (SPINS) A rotation turns a figure around a fixed point. We usually consider the rotation to be clockwise around the origin as the fixed point. Rotation Matrix x Preimage Matrix = New Image Example 6: Rotate the triangle with vertices A(1,1), B (5,2), and C(-2,3) by the stated amount. Give the vertices of the new image. a) 90 degrees b) 180 degrees Example 7: A triangle has vertices A(1,1), B(5,2), and C(-2,3). First, The triangle is rotated 90 degrees. Next, the triangle is reflected across the y-axis. Finally, the triangle is translated right 3 units and down 4 units. What are the FINAL coordinates of the image?