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3/30 Ch. 12A Quiz 4/4 Ch. 12A TEST – part graphing calc + no calc BRING graphing calculators starting TOMORROW! Lesson 12 – 1 Addition of Matrices PRE-CALCULUS Pg. 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29, 35, 37, 39, 43 Learning Objective To add matrices Matrix Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions Dimensions are the rows x columns 2 5 2 𝐵= 1 𝐴 = −1 3 0 2 −7 6 3𝑥2 4𝑥1 Column matrix 𝐶= 6 −3 −1 2 2𝑥2 Square matrix 𝐷= 3 2 1 0 1𝑥4 Row matrix Each number in a matrix is called an element. Matrix We use subscripts to identify position in the matrix, 𝑎𝑖𝑗 5 2 1. In matrix 𝐴 = −1 3 , what is 𝑎32 ? 2 −7 𝑎32 3rd row 2nd column 𝑎32 = −7 Matrix Two matrices are equal iff they have the same dimensions and all of their corresponding elements are equal Matrix Addition Don’t WRITE!! YOU HAVE THIS!! If two matrices, A and B have the same dimensions, then their sum 𝐴 + 𝐵 is a matrix of the same dimensions whose elements are the sums of the corresponding elements of A & B. *Basically match up elements & add Matrix Subtraction Don’t WRITE!! YOU HAVE THIS!! If two matrices, A and B have the same dimensions, then 𝐴 − 𝐵 = 𝐴 + (−𝐵) Properties of Matrix Addition Matrix Don’t WRITE!! YOU HAVE THIS!! If 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are 𝑚 𝑥 𝑛 matrices, then 𝐴 + 𝐵 is an 𝑚 𝑥 𝑛 matrix Closure 𝐴+𝐵 =𝐵+𝐴 Commutative 𝐴 + 𝐵 + 𝐶 = 𝐴 + (𝐵 + 𝐶) Associative There exists a unique 𝑚 𝑥 𝑛 matrix 𝑂 such that 𝑂+𝐴 =𝐴+𝑂 =𝐴 Additive Identity For each 𝐴, there exists a unique matrix −𝐴 such that 𝐴 + −𝐴 = 𝑂 Additive Inverse 5 6 Given: 𝐴 = 1 0 Matrix 7 2 4 2 𝐵= −5 1 1 −4 2. Find 𝐴 + 𝐵 5+4 6+2 1 + (−5) 0 + 1 𝐴+𝐵 = 9 −4 8 8 1 −2 7+1 2 + (−4) 5 6 Given: 𝐴 = 1 0 Matrix 3. Find 𝐴 − 𝐵 5 = 1 7 2 4 2 𝐵= −5 1 = 𝐴 + (−𝐵) −4 −2 −1 6 7 + 5 −1 4 0 2 5 + (−4) 6 + (−2) 7 + (−1) 1+5 2+4 = 1 6 0 + (−1) 4 6 −1 6 1 −4 5 6 Given: 𝐴 = 1 0 Matrix 4. Find 𝐵 − 𝐴 = 4 −5 7 2 = 𝐵 + (−𝐴) 2 1 −5 + 1 −4 −1 4 + (−5) −1 −6 −6 −7 0 −2 2 + (−6) 1 + (−7) −5 + (−1) 1 + 0 = 4 2 𝐵= −5 1 −4 −6 1 −6 −4 + (−2) 1 −4 Properties of Scalar Multiplication Matrix If 𝐴, 𝐵, 𝑎𝑛𝑑 𝑂 are 𝑚 𝑥 𝑛 matrices and 𝑐 and 𝑑 are scalars, then 𝑐𝐴 is an 𝑚 𝑥 𝑛 matrix Don’t WRITE!! YOU HAVE THIS!! 𝑐𝑑 𝐴 = 𝑐(𝑑𝐴) Closure Associative 1∙𝐴 =𝐴 Multiplicative Identity 𝑂𝐴 = 𝑂 and 𝑐𝑂 = 𝑂 Multiplicative Property of the zero scalar & the zero matrix 𝑐 𝐴 + 𝐵 = 𝑐𝐴 + 𝑐𝐵 𝑐 + 𝑑 𝐴 = 𝑐𝐴 + 𝑑𝐴 Distributive Properties Matrices can be used to solve many real world problems. Matrix Change to #5!!!! On half sheet. Not #2. 5. Carl & Flo are training for a triathlon by running, cycling, & swimming. The matrices below show the number of miles that each devotes to each activity, both on weekdays & weekend days. What is the total number of miles that each devotes to each activity in a 7 – day week? Weekend Weekday Carl 6 Run 𝐵 = Cycle 40 Swim 2 Carl 8 Run 𝐴 = Cycle 50 Swim 4 Flo 10 40 2 40 5𝐴 + 2𝐵 = 250 20 12 16 50 200 + 80 90 4 6 10 Total Carl Flo Run 52 66 Cycle = 330 290 Swim 24 16 Flo 8 45 3 Carl: 52 mi run, 330 mi cycle, 24 mi swim Flo: 66 mi run, 290 mi cycle, 16 mi swim You can also solve a “matrix equation” 2 ways (1) Thinking algebraically & treating matrix as a whole Matrix (2) Element by Element 6. Solve 2𝑋 + 3 2 + 3 12 6 1 = 3 6 −2 5 1 4 1 2 = 6 1 −2 5 First distribute the 3 3 6 1 −2 2𝑋 + = 12 3 6 5 Method 1: 1 −2 3 6 2𝑋 = − 6 5 12 3 −2 −8 2𝑋 = −6 2 1 −2 −8 −1 −4 𝑋= 𝑋= 2 −6 2 −3 1 You can also solve a “matrix equation” 2 ways Matrix (2) Element by Element 6. Solve 2𝑋 + 3 2𝑥 2𝑥 2𝑥 3 + 2𝑥 12 6 1 = 3 6 −2 5 1 4 1 2 = 6 1 −2 5 First distribute the 3 3 6 1 −2 2𝑋 + = 12 3 6 5 Method 2: 2𝑥 + 3 = 1 𝑥 = −1 2𝑥 + 12 = 6 𝑥 = −3 2𝑥 + 6 = −2 𝑥 = −4 2𝑥 + 3 = 5 𝑥=1 𝑋= −1 −4 −3 1 Assignment Pg. 600 #4, 5, 8, 10, 14, 16, 19, 20, 23, 25, 27, 29, 35, 37, 39, 43