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. Examples
3x2
2x3
2x2
1x5

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. Example
3x3
3x2
2x3

10. Example
3x3
3x2
2x3

11. Example
3x2
2x3
3x3

12. Example
3x2
2x3
3x3

13. Algorithm
procedure matrix multiplication(A,B:matrices) for i:=1 to m begin for j:=1 to n begin cij := for q := 1 to k cij := cij + aiqbqj end end

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 m begin for j:=1 to n begin cij := for q := 1 to k cij := cij || aiq&&bqj end end

23. Boolean Product =

24. finished