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Matrices Basics

Matrices Basics. Peter Gibby. What’s a Matrix?. A matrix is a way of doing multiple problems at once. Because it works in the same way that a system of equation. A matrix is a rectangular system of elements which can be added or multiplied together.

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Matrices Basics

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  1. Matrices Basics Peter Gibby

  2. What’s a Matrix? • A matrix is a way of doing multiple problems at once. Because it works in the same way that a system of equation. • A matrix is a rectangular system of elements which can be added or multiplied together. • It is first defined with both its dimensions, n x h and the elements. Such as the matrix:

  3. 4 x 3 1 3 -8 2 9 3 A= -3 2 4 3 1 -5

  4. Lets add • Adding matrices requires that they have the same dimensions. • Then each element is added to the corresponding element in the next matrix • Lets remember A, and define B to add them together.

  5. B 7 -6 -2 -4 12 6 B= -10 8 0 -2 6 0

  6. A+B 1+7 3-6 -8-2 2-4 9+12 3+6 -3-10 2+8 4+0 3-2 1+6 -5+0

  7. Answer 8 -3 -10 2 21 9 -13 10 4 1 7 -5

  8. Why Matrices? • We use them when we are given charts of data. For example, each row could represent the amount of money a small business made from selling each product they produce, and the column represents the day • In another chart, we see that the business has to pay employees who each sell their product. Each row represents the employee for each item, and each column the day. • We can subtract these two matrices to find which employees make us the most money each day, and which ones cost the most per day.

  9. Multiplication with a Scalar • When we multiply by a normal number, or “scalar”, we just multiply each element in the matrix by the scalar. • Matrix to matrix multiplication is much more difficult, so I’ll focus on that.

  10. Multiplication Between 2 Matrices • Multiplication doesn’t require the same dimensions, but the first one needs as many columns as the second has rows. • So a 3 x 4 and a 4 x 1 matrix could be multiplied together, but a 3 x 3 and a 2 x 2 cannot be multiplied. Lets define a matrix C and D

  11. C 3 2 2 1 0 8 D 5 7 6 10 12 2

  12. When We Multiply • The new matrix has the number of rows of the first, and the number of columns from the second. For each new value, we multiply the first A value with the first B value, then add it to the second A value times the second B value. This value gets the row number of A and the column number of B And So on for the 3rd and 4th

  13. C x D So we have a 2 x 3 multiplied by a 3 x 2 so our answer will be a 2 x 2 3(5)+2(6)+2(12) 3(7)+2(10)+2(2) 1(5)+0(6)+8(12) 1(7)+0(10)+8(2)

  14. C x D 51 45 101 23

  15. D x C 5(3)+7(1) 5(2)+7(0) 5(2)+7(8) 6(3)+10(1) 6(2)+10(0) 6(2)+10(8) 12(3)+2(1) 12(2)+2(0) 12(2)+2(8)

  16. D x C 22 10 66 28 12 92 38 24 40

  17. Noticing Things • Notice that C x D is not the same as D x C. we have to be careful with matrices because the order we multiply is a huge factor in our answer! • C x D ≠ D x C • (this is kinda super, mega important!)

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