1. Chapter 2 Section 3 Arithmetic Operations on Matrices

2. Matrices Rectangular array of numbers Often described by the size (the number of rows and the number of columns). The matrix 1 2 3 2 3 4 0 8 9 6 7 6 is a 4 x 3 matrix. Four rows and 3 columns

3. Different Types of Matrices See pages 74 – 75 for the definitions of the following matrices: Row Matrix Column Matrix Square Matrix

4. When Two Matrices are Equal Two matrices, A and B, are equal when: 1.They both are the same size AND 2.When ALL the corresponding entries are the equal.

5. Addition and Subtraction of Matrices To add or subtract two matrices together, the two matrices must be exactly the same size. When they are the same size, then add (or subtract) corresponding entries. See the examples at the top of page 76. If the sizes of the two matrices are not the same, then their sum (or difference) is undefined.

6. Multiplication of Matrices A little more complicated than addition and subtraction. In order to multiply matrix A to matrix B (i.e. A · B ) REQUIREMENT: The number of columns in the first matrix (A) must equal the number or rows in the second matrix (B). This is referred to as the inner dimensions being equal.

7. Inner Dimensions Matrices:A · B = C Dimensions: m x n r x c m x c Must be equal 1. If the inner dimensions are not the same, then the product of the two matrices is said to be undefined. 2.If the inner dimensions are the same, then the size of the resulting matrix is the number of rows from the first matrix and the number of columns from the second matrix Size of resulting matrix

8. Example 3(a) (page 80) Calculate the product, if defined. 3– 1 2 0 1 5 1 0 5– 4 2 – 1 Answer: The product is undefined, since the number of columns in the first matrix (2) is not equal to the number of row in the second matrix (3). = Undefined

9. Example 3(b) (page 80) Calculate the product, if defined. 3– 1 2 0 1 5 5 4 – 2 3 Answer: = 3 · 5 + (– 1)·(– 2)3 · 4 + (– 1) · 3 2 · 5 + 0 · (– 2)2 · 4 + 0 · 3 1 · 5 + 5 · (– 2) 1 · 4 + 5 · 3 = 17 9 10 8 – 5 19

10. Example 5 (page 79) Calculate the product using the finger method. 15 3 2 1 2 1 0 Answer: = 1 · 1 + 5 ·11 · 2 + 5 · 0 3 · 1 + 2 · 13 · 2 + 2 · 0 = 6 2 5 6 Intermediate matrix that is not shown in textbook

11. Exercise 21 (page 85) Calculate the product by hand. 3 1 0 2 1 4 3 5 Answer: = 3 · 1 + 1 ·33 · 4 + 1 · 5 0 · 1 + 2 · 30 · 4 + 2 · 5 = 6 17 6 10

12. Exercise 43 (page 85) A = PantsShirtsJackets Mike 6 8 2 Don 2 5 3 B = Pants 20 Shirts 15 Jacket 30 $

13. Exercise 43 (a) AB = 6 8 2 2 5 3 20 15 30 Calculate AB. Solution: = 6 · 20 + 8 · 15 + 2 · 30 2 · 20 + 5 · 15 + 3 · 30 = 340 265

14. Exercise 43 (b) Interpret the entries in AB AB = 340 265

15. Identity Matrix ( I n ) The identity matrix is to matrix multiplication as 1 is to multiplication of numbers. A·I = A I·A = A The identity matrix –Is always a square matrix –Has 1’s on the diagonal (from top left corner to bottom right corner) and 0’s everywhere else.

16. Examples I 2 = I 3 = I 4 = 10 0 1 10 0 0 1 0 0 0 1 10 0 0 0 1 0 0 0 0 1 0 0 0 0 1

17. Matrix Equation Form A system of equations like: – 2x + 4y = 2 – 3x + 7y = 7 can be rewritten as the matrix equation: – 2 4 x 2 – 3 7 y 7 = A = XB ·

18. Matrix Equation Form (continued) Where: –A is the matrix of coefficients –X is the matrix of variables –B is the matrix of constants

19. Exercise 33 (page 85) Give the system of linear equations that is equivalent to the matrix equation: 123x10 456y=11 789z12

20. Exercise 33 (solution) Using the finger method of multiplication: 1x + 2y + 3z10 4x + 5y + 6z=11 7x + 8y + 9z12 x + 2y + 3z = 10 4x + 5y + 6z = 11 7x + 8y + 9z = 12

21. Exercise 37 (page 85) Write the system of linear equations in matrix equation form (i.e. Change it to the form of A X = B): x – 2y + 3z = 5 y + z = 6 z = 2

22. Exercise 37 (solution) 1 – 2 3 x 5 0 1 1 y = 6 0 0 1 z 2 A·X= B