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3.5 Perform Basic Matrix Operations Goal Perform operations with matrices. Your Notes VOCABULARY Matrix A rectangular arrangement of numbers in rows and columns Dimensions The dimensions of a matrix with m rows and n columns are m n. Elements The numbers in a matrix Equal matrices Matrices that have the same dimensions and equal elements in corresponding positions Scalar A real number by which you multiply a matrix Scalar multiplication The process of multiplying each element in a matrix by a scalar ADDING AND SUBTRACTING MATRICES To add or subtract two matrices, simply add or subtract _corresponding_ elements. You can add or subtract matrices only if they have the same _dimensions_. Adding Matrices a b e c d g f a e b f h c g d h Subtracting Matrices a b e c d g f a e b f h c g d h Your Notes Example 1 Add and subtract matrices Perform the indicated operation, if possible. 2 1 1 4 6 2 3 a. 9 3 2 b. 0 3 2 1 7 6 1 5 Solution a. The dimensions of 6 2 are _2 2_ and the dimensions of 3 2 1 3 are _2 1_. So, you _cannot_ subtract the matrices. b. The dimensions are _the same . So, add the matrices. 2 1 1 4 2 1 1 ( 4) 0 3 2 9 0 ( 2) 3 9 7 6 1 5 7 1 6 5 1 3 2 12 8 11 Example 2 Multiply a matrix by a scalar Perform the indicated operation, if possible. a. 3 0 2 7 3(0) 3(2) 3( 7) 1 6 4 3(1) 3(6) 3(4) 0 6 21 12 3 18 1 0 2(1) 2(0) 2 0 b.. 2 3 5 2(3) 2( 5) 6 10 Checkpoint Perform the indicated operation, if possible. 2 3 6 1. 1 5 7 9 2 8 4 1 3 2 2. 3 2 9 6 3 9 6 6 27 18 0 1 4 0 12 0 Your Notes PROPERTIES OF MATRIX OPERATIONS Let A, B, and C be matrices with the same dimensions, and let k be a scalar. Associative Property of Addition (A + B) + C _A (B C)_ Commutative Property of Addition Distributive Property of Addition Distributive Property of Subtraction A B _B A_ k(A B) = _kA kB_ k(A 6) = _kA kB_ Example 3 Solve a multi-step problem Pet Stores Two pet stores sell both dogs and cats. Sales from each store for last month and this month are shown below. Last Month: Store 1 sold 42 dogs, 33 cats. Store 2 sold 56 dogs, 21 cats. This Month: Store 1 sold 36 dogs, 51 cats. Store 2 sold 48 dogs, 37 cats. Organize the data into matrices, then write and interpret a matrix giving the average monthly sales for the two month period. Organize the data into two 2 2 matrices. Last month (A) dogs cats Store 1 42 33 Store 2 56 21 This month(B) dogs cats Store 1 36 51 Store 2 48 37 Write a matrix for the average monthly sales by adding 1 A and B and then multiplying the result by . 2 1 ( A B) 1 42 33 36 51 2 2 56 21 48 37 1 78 84 39 42 2 104 58 52 29 Store 1 sold an average of _39_ dogs and _42_ cats, while Store 2 sold an average of _52_ dogs and _29_ cats. Your Notes Example 4 Solve a matrix equation Solve the matrix equation for x and y. 3x 1 1 3 8 8 4 0 0 6 2 2 y 8 Solution Simplify the left side of the equation. 3 x 1 1 4 0 6 2 3 8 8 2 y 8 0 3x 1 2 8 8 4 2 6 2 y 8 0 8 8 8 12 x 4 8 24 8 y 8 0 Write original equation. Add matrices inside parentheses. Perform scalar multiplication. Equate corresponding elements and solve the two resulting equations. _12x 4_ 8 _24 8y_ 0 x _1_ y _3_ Checkpoint Complete the following exercises. 3. Solve the matrix equation for x and y. 3 2 x 1 2 1 7 5y 6 4 12 4 8 6 x 6, y 1 Homework ________________________________________________________________________ ________________________________________________________________________