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9.2 Using Properties of Matrices

9.2 Using Properties of Matrices. Matrix- a rectangular arrangement of numbers in rows and columns Named with capital letters [ A ] Dimensions- row x column (2 x 4). Element- each number in the matrix Named with lowercase letters and subscripts (a ₂ ₃ ) First number of subscript= row

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9.2 Using Properties of Matrices

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  1. 9.2 Using Properties of Matrices • Matrix- a rectangular arrangement of numbers in rows and columns • Named with capital letters [ A ] • Dimensions- row x column (2 x 4)

  2. Element- each number in the matrix • Named with lowercase letters and subscripts (a₂₃) • First number of subscript= row • Second number of subscript= column

  3. Using Matrices to Represent Points/Figures in Coordinate Planes • Use 2 row matrices to represent the • vertices on a 2-dimensional figure • 1st row = x-coordinate • 2nd row= y-coordinate • Example: Point (2,3)= 2 • 3

  4. Write a matrix to represent the point or polygon. Represent figures using matrices a. Point A b. Quadrilateral ABCD

  5. –4 x-coordinate 0 y-coordinate A B C D –4 –1 4 3 x-coordinates 0 2 1 –1 y-coordinates Represent figures using matrices SOLUTION a. Point matrix for A b. Polygon matrix for ABCD

  6. a. 5 +1 –3 +2 5 –3 1 2 6 –1 + = = 6 +3 –6 + (–4) 6 –6 3 –4 9 –10 b. 1 –7 0 6 – 1 8 –(–7) 5 – 0 6 8 5 5 15 5 – = = 4 –2 3 4 – 4 9 – (– 2) –1 – 3 4 9 –1 0 11 –4 Adding and subtracting matrices

  7. [–3 7]+ [2 –5] 3. =[–1 2 ] [–3 7]+ [2 –5] 4. 1 –4 2 3 – 3 –5 7 8 1 –4 2 3 –1 –7 – = 3 –5 7 8 –4 –13 • Examples: Adding and subtracting matrices In Exercises 3 and 4, add or subtract. SOLUTION SOLUTION

  8. The translation matrix is –1 –1 –1 3 3 3 Represent a translation using matrices 1 5 3 The matrix represents ∆ABC. Find the image 1 0 –1 matrix that represents the translation of ∆ABC1 unit left and 3 units up. Then graph ∆ABCand its image. SOLUTION

  9. A B C A′ B′ C′ –1 –1 –1 1 5 3 0 4 2 + = 3 3 3 1 0 –1 4 3 2 Polygon matrix Image matrix Translation matrix Represent a translation using matrices Add this to the polygon matrix for the preimage to find the image matrix.

  10. 2 –3 1 0 . –1 8 4 5 The matrices are both 2 X 2, so their product is defined. Use the following steps to find the elements of the product matrix. • Multiplying matrices • Number of columns in matrix A= • Number of rows in column B • A X B = A B • (mxn) x (nxp) (m x p) • equal Multiply SOLUTION

  11. 1 0 1(2) + 0(–1) ? 2 –3 = ? ? 4 5 –18 Multiplying matrices STEP 1 Multiply the numbers in the first row of the first matrix by the numbers in the first column of the second matrix. Put the result in the first row, first column of the product matrix.

  12. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = ? ? 4 5 –1 8 Multiplying matrices STEP 2 Multiply the numbers in the first row of the first matrix by the numbers in the second column of the second matrix. Put the result in the first row, second column of the product matrix.

  13. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 4(2) + 5(–1) ? 4 5 –18 Multiplying matrices STEP 3 Multiply the numbers in the second row of the first matrix by the numbers in the first column of the second matrix. Put the result in the second row, first column of the product matrix.

  14. 1 0 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 4(2) + 5(–1) 4(–3) + 5(8) 4 5 –1 8 Multiplying matrices STEP 4 Multiply the numbers in the second row of the first matrix by the numbers in the second column of the second matrix. Put the result in the second row, second column of the product matrix.

  15. 1(2) + 0(–1) 1(–3) + 0(8) 2 –3 = 3 28 4(2) + 5(–1) 4(–3) + 5(8) Multiplying matrices STEP 5 Simplify the product matrix.

  16. –3 B = [2 1] A = ANSWER 4 Yes; the number of columns in Ais equal to the number of rows in B. 6.7 0 = C –9.3 5.2 Multiplying Matrices Use the matrices below. Is the product defined? Explain. 6. AB

  17. –3 B = [2 1] A = 4 ANSWER 6.7 0 Yes; the number of columns in Bis equal to the number of rows in A. = C –9.3 5.2 Multiplying Matrices Use the matrices below. Is the product defined? Explain. 1. BA

  18. –3 B = [2 1] A = ANSWER 4 No; the number of columns in Ais not equal to the number of rows in C. 6.7 0 = C –9.3 5.2 Multiplying Matrices Use the matrices below. Is the product defined? Explain. 2. AC

  19. 3 8 = –4 7 1(3) + 0(–4) 1(8) + 0(7) 3 8 1 0 = 0(3) + 1(–4) 0(8) + 1(7) –4 7 0 1 Examples of Multiplying Matrices Multiply. 3 8 1 0 –4 7 0 1 SOLUTION

  20. Two softball teams submit equipment lists for the season. A bat costs $20, a ball costs $5, and a uniform costs $40. Use matrix multiplication to find the total cost of equipment for each team. Solve a real-world problem SOFTBALL

  21. Solve a real-world problem SOLUTION First, write the equipment lists and the costs per item in matrix form. You will use matrix multiplication, so you need to set up the matrices so that the number of columns of the equipment matrix matches the number of rows of the cost per item matrix.

  22. Dollars Bats Balls Uniforms Dollars 20 Bats ? 13 42 16 Women Women = 5 Balls 15 45 18 ? Men Men 40 Uniforms Solve a real-world problem = EQUIPMENT COST TOTAL COST

  23. 20 13(20) + 42(5) + 16(40) 1110 13 42 16 = = 5 15 45 18 1245 15(20) + 45(5) + 18(40) 40 ANSWER The total cost of equipment for the women’s team is $1110, and the total cost for the men’s team is $1245. Solve a real-world problem You can find the total cost of equipment for each team by multiplying the equipment matrix by the cost per item matrix. The equipment matrix is 2 3 and the cost per item matrix is 3 1, so their product is a 2 1 matrix.

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