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Benn Fox Hannah Weber Column 3 2 4 -1 0 8 Row ! Going vertically is called the column. The column is listed first. Going horizontally is called the row. The row is listed second. For the one above. . .there are 2 rows, so 2 would be listed first. There are 3 columns, so 3 would be listed second. The answer is: 2x3 5 0 4 8 6 5 6 4 9 4 -1 0 8 23 6 7 89 45 35 4 8 41 68 3 Rows This matrix has 3 rows. This matrix has 3 columns. Therefore, the answer would be: 3x3 2 Columns 3 Columns What is the order of the matrices? 7 Rows This matrix has 7 rows. This matrix has 2 columns. Therefore, the answer would be: 7x2 What is the order of the matrices? Example 1 5 0 4 9 7 6 5 6 4 2 4 8 Example 2 5 0 3 3 5 6 9 7 6 5 6 4 2 4 8 6 Columns 4 2 Rows Answer 2 5 0 3 3 1 Row 5 6 6 Columns Answer 1 5 0 There are 2 rows. There are 6 columns Therefore, the answer is: 2x6 There is 1 rows There are 6 columns Therefore, the answer is: 1x6 a11 a12 a13 . . . . a1n a21 a22 a23 . . . . a2n am3 .... am2 .... .... .... am1 amn An m x n matrix is a rectangular array of m rows and n columns of real numbers. The first subscript number identifies the row it’s in. The second subscript number identifies which column it’s in. -2 -6 -4 -5 4 7 9 8 9 7 6 5 Because the first letter is 1, that means it’s in the first row. For an example: Identify the element specified for the following matrix. a13 Because the second letter is 3, that means it’s in the third column. -2 -6 -4 -5 -2 -6 -4 -5 4 7 9 8 4 7 9 8 9 7 6 5 9 7 6 5 -2 -6 -4 -5 4 7 9 8 9 7 6 5 Because it’s in the 1st row, and the 3rd column, the answer would be: -4 Identify the element specified for the following matrix. a21 -2 -6 4 7 9 7 It’s in the 2nd row. It’s in the 1st column. Because it’s in the 2nd row, and the 1st column, the answer would be: 4 Example 1 Identify the element specified for the following matrix: a44 Example 2 8 9 7 -4 Identify the element specified for the following matrix: a15 5 8 2 0 43 5 7 -4 8 4 12 3 25 5 4 322 0 3 2 6 8 4 5 45 3 25 6 2 3 8 9 3 543 8 4 55 Answer 1 Identify the element specified for the following matrix: a44 Answer 2 8 9 7 -4 Identify the element specified for the following matrix: a15 5 8 2 0 43 5 7 -4 8 4 12 3 25 5 4 322 0 3 2 6 8 4 5 45 3 25 6 2 3 8 9 3 543 8 4 55 Because it’s in the 4th row. Because it’s in the 4th column. The answer would be: 4 Because it’s in the 1st row. Because it’s in the 4th column. The answer would be: 8 To add or subtract matrices they need to have: The same sized rows The same sized columns 7 + 6 = 13 7 2 9 4 ! 6 1 1 0 13 3 10 4 Add or subtract the numbers in the matching positions. 2 6 4 9 8 12 5 15 4 6 9 18 2+5 = 7, 4+4=8, 8+9=17 6+15=21, 9+6=15, 12+18=30 9 3 5 4 2 4 0 4 1 9-0=9, 4-7=-3 3-4=-1, 2-1=1 5-1=4, 4-3=1 7 1 3 7 21 8 15 17 30 Example 1 4 7 10 Example 2 5 3 5 8 11 8 12 6 9 12 15 14 3 2 1 6 4 5 1 33 18 13 9 6 10 1 10 4 7 10 Answer 1 5 8 11 3 2 1 6 9 12 6 4 5 9 6 10 4+3=7, 5+6=11, 6+9=15 7+2=9, 8+4=12, 9+6=15 10+1=11, 11+5=16, 12+10=22 Answer 2 5 3 8 12 15 14 1 33 18 13 1 10 5-1=4, 8-18=-10, 15-1=14 3-33=-30, 12-13=-1, 1410=4 4 −30 7 9 11 −10 −1 11 12 16 14 4 15 15 22 ! Columns of the first matrix equals the number of rows in the second matrix 4 5 3 4 6 6 2x3 9 2 6 4 3 4 3x2 The same numbers, mean you can multiple. ! The outer numbers show what dimensions the answer will be. ! 1. 2. 3. Make sure the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix. Multiply the numbers of each row of the 1st matrix with the numbers of each column in the second matrix. Add up the products from step 2. 1 0 2 3 1 −1 2x3 1 0 2 1 0 −1 3x2 Multiply each number from the first row in the 1st matrix with the 1st column of the 2nd matrix, then add them up. 1(1) 1(0) 0(1) 0(0) + + + + 2(2) 2(1) 1(2) 1(1) + + + + 3(0) = 5 3(-1) = -1 -1(0) = 2 -1(-1) = 2 Multiply each number from the first row in the 1st matrix with the 2nd column of the 2nd matrix, then add them up. Repeat these steps with the 2nd row of the 1st matrix. Answer = 5 −1 2 2 1 0 2 1 0 −1 1(1) 1(2) 1(3) 2(1) 2(2) 2(3) 0(1) 0(2) 0(3) + + + + + + + + + 1 2 0 1 0 2 3 1 −1 0(0) = 1 0(1) = 2 0(-1) = 3 1(0) = 2 1(1) = 5 1(-1) = 5 -1(0) = 0 -1(1) = -1 -1(-1) = 1 2 3 5 5 −1 1 0 −6 2 −2 4 −6 4 5 −2 0 6 −6 0(4) + 2(-2) + -2(6) = -16 0(5) + 2(0) + -2(-6) = 12 -6(4) + 4(-2) + -6(6) = -68 -6(5) + 4(0) + -6(-6) = 6 −16 12 −68 6 1 30 2 3 1 −1 You distribute the 3 to each of the numbers. 3 0 6 9 3 −3 Example 1 −1 3 2 4 Example 2 0 0 1 0 1 0 1 0 0 Example 3 3 4 6 4 4 5 7 5 6 8 −3 5 1 2 1 2 0 1 −1 3 4 Answer 1 −1 3 Answer 2 0 0 1 0 1 0 1 0 0 Answer 3 3 4 4 5 2 4 4 6 5 7 6 8 −3 The first matrix is a 2 x 2. The second matrix is a 1 x 2. Because of this, the answer would be: Not Possible 5 1 2 1 2 0 1 −1 3 4 12 16 20 16 24 20 28 24 32 0(1) 0(2) 0(1) 0(1) 0(2) 0(1) 1(1) 1(2) 1(1) + + + + + + + + + 0(2) 0(0) 0(1) 1(2) 1(0) 1(1) 0(2) 0(0) 0(1) + + + + + + + + + 1(-1) = -1 1(3) = 3 1(4) = 4 0(-1) = 2 0(3) = 0 0(4) = 1 0(-1) = 1 0(3) = 2 0(4) = 1 −1 3 4 2 0 1 1 2 1 At a zoo, kids ride a train for 25 cents. Adults ride it for $1. Senior citizens for 75 cents. On a given day: 1,400 paid a total of $740 for the rides. There were 250 more kids than all other riders. Find the total amount of children, adults, and senior citizens. 1st step: assign letters for each variable. x=children y=adults z=senior citizens 2nd step: set up equations. .25x+y+.75z=740 x+y+z=1400 x-(y+z)=250 .25 for each kid, 1 dollar for each adult, .75 for each senior citizen. All three of the variables = 1,400 total paid 250 more kids than all other riders. 3rd Step: Plug into calculator as a matrix 4th Step: Find inverse of with the calculator 5th Step: Multiply the answer from the 4th step with Answer: Example 1 Matt has 74 coins: nickels, dimes, and quarters. For a total of $8.85. The number of nickels and quarters is 4 more than the number of dimes. Find the number of each coin. Answer 1 Matt has 74 coins: nickels, dimes, and quarters. For a total of $8.85. The number of nickels and quarters is 4 more than the number of dimes. Find the number of each coin. N + D+ Q =74 .05N +.10D + .25Q = 8.85 N- D+ Q = 4 When a nxn matrix with 1’s on the main diagonal and 0’s everywhere else, it is considered an identity matrix. When you multiply it with another matrix, the answer will come out the same. To find the inverse of a matrix: 4 2 7 6 -1 -1 1 − − − 1 (4 ∗ 6) − (7 ∗ 2) 1 10 6 −7 −2 4 6 −7 −2 4 .6 −.7 −.2 .4 Example 1 Find the inverse of : 8 −5 −3 2 Example 2 Find the inverse of : 5 16 −1 −3 Answer 1 Find the inverse of : 8 −5 −3 2 Answer 2 Find the inverse of : 5 16 −1 −3 2 3 5 8