1. Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5th Edition) Chapter 2 The Fundamentals: Algorithms, the Integers, and Matrices

2. Section 2.7 Matrices

3. Matrix Definition A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an mxn matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.

4. 2x3 3x2 1x5 2x2 Examples

5. Sum of two matrices Let A = [aij] and B = [bij] be mxn matrices. The sum of A and B, is the mxn matrix that has aij + bij as its (i,j)th element. In other words, A + B = [aij + bij].

6. Examples

7. Examples Not Possible

8. Matrix Multiplication Let A be an mxk matrix and B be a kxn matrix. The product of A and B, denoted by AB, is the mxn matrix with (i,j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij], then

9. 3x2 2x3 3x3 Example

10. 3x2 2x3 3x3 Example

11. 3x2 2x3 Example 3x3

12. 3x2 2x3 Example 3x3

13. Algorithm procedure matrix multiplication(A,B:matrices)for i:=1 to mbeginfor j:=1 to nbegin cij := 0for q := 1 to k cij := cij + aiqbqjendend

14. Identity Matrix Exist as square matrices. AI = IA = A

15. Transpose of a matrix The transpose of a square matrix A is labeled AT and is found by interchanging rows and columns.

16. Symmetric Matrices A square matrix A is called symmetric if A = AT.

17. Zero-one Matrices

18. Join of two zero-one matrices

19. Join

20. Meet of two zero-one matrices

21. Boolean Product

22. Algorithm procedure matrix booleanProduct(A,B:matrices)for i:=1 to mbeginfor j:=1 to nbegin cij := 0for q := 1 to k cij := cij|| aiq&&bqjendend

23. Boolean Product =

24. finished