Matrix Operations (Adding, Subtracting, and Multiplying) Ch. 3.5, 6 Finding the Determinant and Inverse of a Matrix Ch. 3.7, 8

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1 Matrix Operations (Adding, Subtracting, and Multiplying) Ch. 3.5, 6 Finding the Determinant and Inverse of a Matrix Ch. 3.7, 8 Objectives: Add, subtract, and multiply matrices. Find the determinant and inverse of a matrix.

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3 Matrix Operations Ch. 3, Sec. 5 Matrix rectangular arrangement of numbers in rows and columns. The order of matrix is always rows and columns. ex. L M N O Q P 2 x 3 A row matrix is a matrix that has only one row. A column matrix is a matrix with only one column. A square matrix has the same number of rows and columns.

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5 EXAMPLE 1 Add and subtract matrices Perform the indicated operation, if possible a b

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7 EXAMPLE 2 Multiply a matrix by a scalar Perform the indicated operation, if possible. 4 1 a b

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9 EXAMPLE 3 Solve a multi-step problem

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11 EXAMPLE 4 Solve a matrix equation Solve the matrix equation for x and y. 5x y

12 Matrix Multiplication Chapter 3, Sec. 6 Matrix multiplication of A and B is defined only if the number of columns of A equals the number of rows of B. Example 1 A L M MM N O -1 3 P L PP B M O and N P -4 1 Q 5 0 Q L M MM N O P PP Q L O M -3 2 N P -4 1Q L M MM N O P PP Q The number of rows of A determines the number of rows of the solution and the columns of B determines the number of columns of the solution.

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14 Example 2 Finding the Product of Two Matrices a. L N M O L P M Q MM N O P PP Q L O M N P Q b. L N M L O M N P -2 5Q O 3 4 P L Q M O N P 0 1 Q 3 4 c. d. L O M 2 MM -1 N P PP Q L O M 2 MM -1 N 1 P PP Q L M MM N O P PP Q

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17 EXAMPLE 3 Describe matrix products State whether the product AB is defined. If so, give the dimensions of AB. a. A: 4 3, B: 3 2 b. A: 3 4, B: 3 2

18 EXAMPLE 4 Find the product of two matrices Find AB if A = and B =

19 EXAMPLE 5 Use matrix operations Using the given matrices, evaluate the expression. A = , B = , C = a. A(B + C) b. AB + AC

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21 EXAMPLE 6 Use matrices to calculate total cost Sports Two hockey teams submit equipment lists for the season as shown. Each stick costs $60, each puck costs $2, and each uniform costs $35. Use matrix multiplication to find the total cost of equipment for each team.

22 Determinants Ch. 3.7 Objectives: Find the determinant of a matrix Use determinant to solve problems involving the area of a triangle.

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24 EXAMPLE 1 Evaluate determinants Evaluate the determinant of the matrix. a SOLUTION b

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26 EXAMPLE 2 Find the area of a triangular region Sea Lions Off the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Año Nuevo Island, as shown. (In the map, the coordinates are measured in miles.) Use a determinant to estimate the area of the region. x1 y1 1 1 Area tri. ± x2 y2 1 2 x y 1 3 3

27 EXAMPLE 3 Use Cramer s rule for a 2 2 system Use Cramer s rule to solve this system: 9x + 4y = 6 3x 5y = 21 Just solve the system using elimination, substitution, matrix (rref( )), or graphing.

28 EXAMPLE 4 Solve a multi-step problem CHEMISTRY The atomic weights of three compounds are shown. Use a linear system and Cramer s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O).

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30 Finding the Inverse of a Matrix Ch. 3.8 Objective: Use the inverse of a matrix to solve linear systems

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33 Recall

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36 EXAMPLE 1 Find the inverse of a 2 2 matrix Find the inverse of A =

37 EXAMPLE 2 Solve a matrix equation Solve the matrix equation AX = B for the 2 2 matrix X. A B X = X A B

38 EXAMPLE 3 Find the inverse of a 3 3 matrix Use a graphing calculator to find the inverse of A.Then use the calculator to verify your result. A = Enter matrix A into a graphing calculator and calculate A 1. Then compute AA 1 and A 1 A to verify that you obtain the 3 3 identity matrix.

39 EXAMPLE 3 Find the inverse of a 3 3 matrix

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41 EXAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2x 3y = 19 x + 4y = 7 Equation 1 Equation 2 coefficient matrix of matrix of matrix (A) variables (X) constants(b) x y = 19 7 x y A 1 B

42 EXAMPLE 5 Solve a multi-step problem Gifts A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $ A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $ Find the cost of each item in the gift baskets.

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47 Lesson Close: What did we discuss today? Did I already know some / all of the information? How might I use this outside of school?

48 Assignment: p. 191 >> even p. 200 >> 2-34 even, 38, 40 p. 208 >> 4-20 even, 40 p. 214 >> 4-10 even, 11, 14, 18, 20, 22, 24, 51, 52, 53 p. 219 >> 2, 3, 4