I. Matrix Logic A. Definition

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1 Section 4. and 4.2 An Introduction to Matrices and Adding Matrices To solve problems using matrix logic. To create a matrix and name it using dimensions. 3. To perform scalar multiplication on a matrix 4. To find unknown values in equal matrices. 5. To add and subtract matrices. I. Matrix Logic A. Definition B. Examples Jim, Tom and Mike are married to Brenda, Kelly and Lisa. Use these clues to find out who is married to whom. Tom is Kelly s brother and lives in Florida Mike is shorter than Lisa s husband. 3. Mike works at a bank. 4. Brenda and her husband live in Kentucky. 5. Kelly and her husband work in their candy store. II. Matrices A. Definitions Matrix a of and _ ( elements so that in the matrix has a ) Element each or of a the matrix 3. Dimensions tells how many _ and a matrix has. T m = rows and n = columns m n 4. Square Matrix number of _ and 5. Equal Matrices same and their are. B. Examples 2x + y 6 = x 3y 3 5x y 3 8a 54 = 4 27b III. Matrix Operations (Part A) A. Matrix Addition and Subtraction Note: Matrices can only be added if they are the same dimension Example: B. Scalar Multiplication Homework: p all, 6-24 evens, 26, 30, 35, 36 and p odds, all, 42, 43

2 Section 4.3 Multiplying Matrices (Matrix Operation Part B) To multiply two matrices Note: m x n matrix can only be by a n x r matrix. The result is a m x r matrix (i.e., the number _ of the matrix must the number of in the second matrix) Determine whether each matrix product is defined. If so, state the dimensions of the product. A3 x4 and B 4x2 A3 x2 and B 4x3 Examples: If 3 2 A = 2 and 5 2 B = 4 3 Find AB Find BA 3. On two days the student store sold the following amounts of pencils, erasers, and binders Pencils Erasers Binders Monday Tuesday If pencils sold for $0.20, erasers for $0.35 and binders for $85, set up a matrix multiplication problem to find the total sales for each day. Homework: p all, 9-25 odds, 29, 30, 3-34 all, all, all

3 Section 4.5 Matrices and Determinants To evaluate the determinant of a 2 x 2 and a 3 x 3 matrix. To find the area of a triangle given the coordinates of its vertex.. I. Determinants A. 2 x 2 a b ad cb c d = a b c a b c a b B. 3 x 3 (wrap around method) d e f = d e f d e = ( aei + bfg + cdh) ( gec + hfa + idb) g h i g h i g h C. Examples x 3 2x = II. Area of a Triangle A. Formula If given the vertices of a triangle, ( a, b),( c, d), and ( e, f ), then a b A = c d 2 e f B. Example: Find the area of a triangle whose vertices have coordinates: (,-), (4,3), and (0,5). Homework: p. 85 2, 3, 4, (5-24)/3, odds, 40, 4, all, all

4 Section 4.6 Cramer s Rule To use Cramer s rule to solve systems of linear equations I. Cramer s Rule C. Rule: D. If If ax + b y = c a x + b y = c, then a x + b y + c z = d a x + b y + c z = d a x + b y + c z = d E. Examples 3x 2y = 24 5x + y = 4, then x = and y = x =, y =, z = 2x y + z = 2 x + 2y + 6z = 3 3x y + 2z = 3. At the Taco Waco, Carmen bought 3 tacos, 3 burritos and 3 drinks for $6.0. Rosemary bought 2 tacos, 5 burritos and 4 drinks for $8.63. Jackie bought 4 tacos, 5 burritos and 4 drinks for $6.00. What is the cost of each?

5 III. Test for a unique solution A. If the does _ equal zero, then the system has a solution. B. Note: If the coefficient determinate zero, then either or _ solutions. No Solution occurs when the coefficient determinate equals zero. Infinitely many solutions occur when coefficient determinate equals zero and of the numerator determinates equals zero. A. Examples: Determine whether each system has a unique solution. 2x y + z = 2 6x + 7 y = 9 x + 2y + 6z = x y = 5 3x y + 2z = Homework: p. 92 3, (2-24)/3, 27-3 odds, all, all, 48-5 all

6 Section 4.7 Identity and Inverse Matrices To write the identity matrix for any matrix. To find the inverse matrix of a 2 x 2 matrix. I. Identity Matrix A. Def The identity matrix is a matrix that, when _ by another, the same matrix. B. C. Find the identity matrix for each [ 6 2 5] II. Inverse Matrix for 2 x 2 matrices A. Theorem: If matrix A has an inverse matrix A -, then A A = A A = I. 5 3 B. Example: If A = 2, find A - a b 5 3 a b 0 Let A = c d, then A A = I. (i.e., 2 = c d 0 ) C. Inverse Matrix for 2 x 2 a b d b If M = c d then M =, M 0 M c a. Example: Find the inverse of D. Inverses for all other square matrices can be done on the calculator. Example: Homework: p. 98,-5 odds, 6-9 all, 2-3 odds, 45-7 all

7 Section 4.8 Using Inverse Matrices To write a system of linear equations as a matrix and use inverses to solve. How to: Examples Write as a matrix equation: 4 x 2 y = 7 x + 6y = 9 x 2y + z = 4 Write as a matrix equation: 3x 3y + 5z = 22 x + 4y 7z = Solve using matrix equations: 7x y = 0 3x + 2y = Solve using matrix equations: 8x y + 2z = 7 5x + y z = 2 4x 2y + 4z = 2 5. Dan bought 6 pieces of gum and 7 jawbreakers for $0.76. Tom bought 7 pieces of gum and 4 jawbreakers for $0.60. What was the cost of each? Use matrix equations to solve. Homework: p. 205, 3, (2-30)/3, 33, 34, 39-4 all, 42-5 all

A2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems. DUE Date: Friday 12/2 as a Packet. 3x 2y = 10 5x + 3y = 4. determinant.

A2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems Name: DUE Date: Friday 12/2 as a Packet What is the Cramer s Rule used for? à Another method to solve systems that uses matrices and determinants.