1. CSNB143 – Discrete Structure Topic 3 – Matrices

2. Learning Outcomes Students should understand all matrices operations. Students should be able to differentiate different type of matrices and operations by different matrix. Students should be able to identify Boolean matrices and how to operate them.

3. Topic 3 – Matrices An array of numbers arranged in m horizontal rows and n vertical columns. We say that A is a matrix m x n. (Dimension of matrix).

4. Topic 3 – Matrices Square Matrix Number of rows = number of columns Which one(s)of the following is(are) square matrix(ces)? Where is the main diagonal?

5. Topic 3 – Matrices Diagonal Matrix “a square matrix in which entries outside the main diagonal area are all zero, the diagonal entries may or may not be zero”

6. Topic 3 – Matrices Equal Matrix Matrices are equal if the corresponding elements are equal Example: A and B are equal matrices, find the values of a, b, x and y

7. Topic 3 – Matrices Equal Matrices - Work this out 1.If 2. If Find a, b, c, and d Find a, b, c, k, m, x, y, and z

8. Topic 3 – Matrices Matrices Summation The sum of the matrices A and B is defined only when A and B have the same number of rows and the same number of columns (same dimension)

9. Topic 3 – Matrices Matrices Summation – work this out a) Identify the pair of which matrices between which the summation process can be executed b) Compute C + G, A + D, E + H, A + F.

10. Topic 3 – Matrices Matrices Products Steps before 1.Find out if it is possible to get the products? 2.Find out the result’s dimension 3.Arrange the numbers in an easy way to compute – avoid confusion

11. Topic 3 – Matrices Matrices Products – Possible outcomes

12. Topic 3 – Matrices Matrices Products – Work this out Let Show that AB is NOT BA

13. Topic 3 – Matrices Transposition Matrix A matrix which is formed by turning all the rows of a given matrix into columns and vice-versa. The transpose of matrix A is written A T.

14. Topic 3 – Matrices Transposition Matrix – Work this out Compute (BA) T : Compute AT (D + F)

15. Topic 3 – Matrices Symmetrical Matrix A is said to be symmetric if all entries are symmetrical to its main diagonal.

16. Topic 3 – Matrices Boolean Matrix and Its Operations Boolean matrix is an m x n matrix where all of its entries are either 1 or 0 only. There are three operations on Boolean: – Join by – Meet – Boolean Product

17. Topic 3 – Matrices Boolean Matrix and Its Operations – Join By Given A = [a ij ] and B = [b ij ] are Boolean matrices with the same dimension, join by A and B, written as A  B, will produce a matrix C = [c ij ], where c ij = 1if a ij = 1 OR b ij = 1 0if a ij = 0 AND b ij = 0

18. Topic 3 – Matrices Boolean Matrix and Its Operations – Meet Meet for A and B, both with the same dimension, written as A  B, will produce matrix D = [d ij ] where d ij = 1if a ij = 1 AND b ij = 1 0 if a ij = 0 OR b ij = 0

19. Topic 3 – Matrices Boolean Matrix and Its Operations – Boolean Products If A = [a ij ] is an m x p Boolean matrix, and B = [b ij ] is a p x n Boolean matrix, we can get a Boolean product for A and B written as A ⊙ B, producing C, where:

20. Topic 3 – Matrices

21. Boolean Matrices – work this out