1. Matrix and Matrices

2. Matrix Notation It is used to represent multiple data Your work company runs two types of shop, whole-sale and retail. Your company locates in three geographical regions. North: 20 retail and 2 wholesale Center: 20 retail and 2 wholesale South: 20 retail and 2 wholesale

3. Matrix Notation You can represent this company as: Type of shop RetailWholesale RegionNorth202 Center606 South304

4. Matrix Notation Simply: A= 202 604 306 Matrix name Capital Dimension: 3 X 2 R X C Elements: a 1,2 = 2, a3,1 = 30 a 3,2 = ??? One row matrix called Row Victor One column matrix called Column Victor

5. Example A company makes three products from four raw materials. It uses: Product1: 17,22,4,7 Product2: 12,5,22,6 Product3: 7,13,14,8 Show data as matrix, M. What are the values of m 2,3 and m 3,2

6. Continued Table notation: Matrix notation: M = Raw Material 1 Raw Material 2 Raw Material 3 Raw Material 4 Product1172247 Product2125226 Product373148 172247 125226 73148 m 2,3 = 22, m 3,2 = 3

7. Transpose Matrix The transpose matrix of A is written as A t: A = A t = ab cd ef ace bdf

8. Matrices Mathematics ( +, - ) 257 463 612 08 A= B= C = A + B, then C = 869 4511 D = A - B, then D = -445 45

9. Matrices Mathematics( x ) To multiply number by matrix, multiply this number in every element in the matrix. Example: M = Then, 3 x M = 32 7-5 96 21-15

10. Matrices Mathematics( x ) To multiply two matrices A and B, then the number of column in matrix A must equal the number of rows in matrix B. Example: A = B= Find D = A x B D = = 31 25 -20 7-31 432 3x7+1x43x-3+1x33x1+1x2 2x7+5x42x-3+5x32x1+5x2 -2x7+0x4-2x-3+0x3-2x1+0x2 25-65 34910 -146-2

11. Identity Matrix Identity matrix I is the matrix that contains the following elements: I = x = 10 01 A x I = I x A = A 32 7-5 10 01 32 7

12. Matrix Inversion If we have matrix A, its inverse is called A -1 A x A -1 = A -1 x A = I Matrix must be square Matrix division is not defined. Exercise: How can you prove that the inverse of the matrix A = is A -1 = 15 26 -1.51.25 0.5-0.25

13. Using Matrices to solve Simultaneous Equations If we have A x X = B, X is unknown matrix, then we can slove this equation as the following: A x X = B A -1 x A x X = A -1 x B I x X = A -1 x B X = A -1 x B

14. Continued Example: Use Matrix inversion to solve the simultaneous equations: 4x + y = 13 3x + 2y = 16 A = X = B = X = A-1 x B X = x = = 41 32 x y 13 16 0.4-0.2 -0.60.8 13 16 2 5 x y