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7.2 Matrix Algebra

7.2 Matrix Algebra. Benn Fox Hannah Weber. Order of Matrices. Column. Row. !. Going vertically is called the column. The column is listed first. Going horizontally is called the row. The row is listed second.

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7.2 Matrix Algebra

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  1. 7.2 Matrix Algebra Benn Fox Hannah Weber

  2. Order of Matrices Column 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: 2 x 3

  3. Orders of Matrices Continued What is the order of the matrices? 3 Columns 2 Columns 3 Rows This matrix has 3 rows. This matrix has 3 columns. Therefore, the answer would be: 3 x 3 7 Rows This matrix has 7 rows. This matrix has 2 columns. Therefore, the answer would be: 7 x 2

  4. Order of Matrices Examples What is the order of the matrices? Example 1 Example 2

  5. Order of Matrices Answers There are 2 rows. There are 6 columns Therefore, the answer is: 2 x 6 6 Columns Answer 1 2 Rows There is 1 rows There are 6 columns Therefore, the answer is: 1 x 6 6 Columns Answer 2 1 Row

  6. Identifying the Element Specified . . . . . . . . . . . . . . . . . . . . . . . . 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.

  7. Identifying the Element Specified For an example: Identify the element specified for the following matrix. a13 Because the first letter is 1, that means it’s in the first row. Because the second letter is 3, that means it’s in the third column.

  8. Identifying the Element Specified Continued Identify the element specified for the following matrix. a21 • It’s in the 2nd row. • It’s in the 1st column. Because it’s in the 1st row, and the 3rd column, the answer would be: -4 Because it’s in the 2nd row, and the 1st column, the answer would be: 4

  9. Identifying the Element Specified Examples Example 1 Identify the element specified for the following matrix: a44 Example 2 Identify the element specified for the following matrix: a15

  10. Identifying the Element Specified Answers Answer 1 Identify the element specified for the following matrix: a44 Answer 2 Identify the element specified for the following matrix: a15 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

  11. Adding and Subtracting Matrices To add or subtract matrices they need to have: • The same sized rows • The same sized columns 7 + 6 = 13 = ! Add or subtract the numbers in the matching positions.

  12. Adding and Subtracting Continued = 2+5 = 7, 4+4=8, 8+9=17 6+15=21, 9+6=15, 12+18=30 = 9-0=9, 4-7=-3 3-4=-1, 2-1=1 5-1=4, 4-3=1

  13. Adding and Subtracting Examples ? = Example 1 ? = Example 2

  14. Adding and Subtracting Answer = Answer 1 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 5-1=4, 8-18=-10, 15-1=14 3-33=-30, 12-13=-1, 14-10=4

  15. Multiplying Matrices Columns of the first matrix equals the number of rows in the second matrix ! ! ! The outer numbers show what dimensions the answer will be. 2 x 3 3 x 2 The same numbers, mean you can multiple.

  16. How to Multiply • 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.

  17. Multiplying Matrices 1(1) + 2(2) + 3(0) = 5 1(0) + 2(1) + 3(-1) = -1 0(1) + 1(2) + -1(0) = 2 0(0) + 1(1) + -1(-1) = 2 2 x 3 3 x 2 1. Multiply each number from the first row in the 1st matrix with the 1st column of the 2nd matrix, then add them up. 2. Multiply each number from the first row in the 1st matrix with the 2ndcolumn of the 2nd matrix, then add them up. 3. Repeat these steps with the 2nd row of the 1st matrix. Answer =

  18. Multiplying Matrices Continued 1(1) + 0(0) = 1 1(2) + 0(1) = 2 1(3) + 0(-1) = 3 2(1) + 1(0) = 2 2(2) + 1(1) = 5 2(3) + 1(-1) = 5 0(1) + -1(0) = 0 0(2) + -1(1) = -1 0(3) + -1(-1) = 1 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 = 3 You distribute the 3 to each of the numbers.

  19. Multiplying Matrices Examples = ? Example 1 = ? Example 2 = ? Example 3 4

  20. Multiplying Matrices Answers 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 = Answer 1 = 0(1) + 0(2) + 1(-1) = -1 0(2) + 0(0) + 1(3) = 3 0(1) + 0(1) + 1(4) = 4 0(1) + 1(2) + 0(-1) = 2 0(2) + 1(0) + 0(3) = 0 0(1) + 1(1) + 0(4) = 1 1(1) + 0(2) + 0(-1) = 1 1(2) + 0(0) + 0(3) = 2 1(1) + 0(1) + 0(4) = 1 Answer 2 = Answer 3 4

  21. Matrix Applications • 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.

  22. Matrix Applications Continued • 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:

  23. Matrix Application Example • 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. Example 1

  24. Matrix Application Example 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 =

  25. Identity Matrix and Inverses • 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. =

  26. Inverse of a Matrix • To find the inverse of a matrix: -1 = -1 = = =

  27. Inverse of Matrices Examples Example 1 Find the inverse of : Example 2 Find the inverse of :

  28. Inverse of Matrices Answers Answer 1 = = Find the inverse of : Answer 2 = = Find the inverse of :

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