1. CS100: Discrete structures
Computer Science Department
Lecture 5 - Matrices

2. Matrices Introduction
DEFINITION 1:
A matrix is a rectangular array of numbers.
A matrix with m rows and n columns is called an m x n matrix.
The plural of matrix is matrices.
A matrix with the same number of rows as columns is called square.
Two matrices are equals if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.
Example: The matrix is a 3 x 2 matrix.
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3. Matrices
DEFINITION 2:
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4. Example 1
Then A is 2 x 3 with a12 = 3 and a23 = 2, B is 2 x 2 with b21 = 4, C is 1 x 4,
D is 3 x 1, and E is 3 x 3 ? ? ? ? ? ? ? ?
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5. Exercise 3×4 Let What size is A ? What is the third column of A ?
What is the second row of A ?
What is the element of A in the (3,2)th position ?
3×4
A(3,2)=1
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6. Diagonal Matrix Example :
A square matrix A = [aij ] for which every entry off the main diagonal is zero, that is, aij = 0 for i ≠ j, is called a diagonal matrix
Example :
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7. Identity Matrix The n x n diagonal matrix
all of whose diagonal elements are 1, is called the identity matrix of order n.
Multiplying a matrix by an appropriately sized identity matrix does not change this matrix. In other words, when A is an m x n matrix, we have
AIn= ImA = A
Powers of square matrices can be defined. When A is an n x n matrix, we have
A0 = In , Ar = AAAA…A (r times)
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8. Example of Matrix applications
Matrices are used in many applications in computer science, and we shall see them in our study of relations and graphs.
At this point, we present the following simple application showing how matrices can be used to display data in a tabular form
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9. Cont’d
The following matrix gives the airline distance between the cities indicated
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10. Equal Matrices if aij = bij , DEFINITION 2:
Two m x n matrices A = [aij ] and B = [bij] are said to be equal
if aij = bij ,
1 ≤ i ≤ m , 1 ≤ j ≤ n; that is, if corresponding elements are the same.
Notice how easy it is to state the definition using generic elements aij , bij
Two matrices are equal if they have the same dimension or order and the corresponding elements are identical.
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11. Then A = B if and only if X=-3, y=0, and z=6
Cont’d
Then A = B if and only if X=-3, y=0, and z=6
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12. Matrices Matrix Arithmetic
DEFINITION 3:
Let A = [aij] and B = [bij] be m x n matrices.
The sum of A and B, denoted by A + B, is the m x n matrix that has aij + bij as its (i,j)th element.
In other words, A + B = [aij + bij].
The sum of two matrices of the same size is obtained by adding elements in the corresponding positions.
Matrices of different sizes can’t be added.
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13. Example 1
Example 2
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14. Zero Matrix
A matrix all of whose entries are zero is called a zero matrix and is denoted by 0
Each of the following is Zero matrix
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15. Properties of Matrix Addition
A + B = B + A
(A + B) + C = A + (B + C)
A + 0 = 0 + A = A
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16. Matrices Production DEFINITION 4:
Let A be an m x k matrix and B be a k x n matrix.
The product of A and B, denoted by AB, is the m x n matrix with its (i,j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the ith column of B. In other words, if AB = [cij], then
cij = ai1b1j + ai2b2j + … + aikbkj.
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17. Matrices Production
The product of the two matrices is not defined when the number of columns in the first matrix and the number of rows in the second matrix is not the same.
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18. Matrices Production
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19. AB=
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20. Matrices Production Example: Find AB if it is defined.
Let A 4X3= and B3X2 =
Find AB if it is defined.
AB4X2 =
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21. Exercise
Consider the matrices A= ,B= ,C= Find the following: 2A= 4A + B = A+0=
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22. Matrices Production No Solution: Example: Let A 2x2 = and B 2x2=
DEFINITION 5:
If A and B are two matrices, it is not necessarily true that AB and BA are the same.
E.g. if A is 2 x 3 and B is 3 x 4, then AB is defined and is 2 x 4, but BA is not defined. Even when A and B are both n x n matrices, AB and BA are not necessarily equal.
Example:
Let A 2x2 = and B 2x2=
Does AB = BA?
Solution:
AB = and BA = No
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23. Properties of Multiplication
If A = m x p matrix, and B is a p x n matrix, then AB can be computed and is an m x n matrix.
As for BA, we have four different possibilities:
BA may not be defined; we may have n ≠ m
For Example : A=4x5 , B=5x6
BA may be defined if n = m, and then BA is p x p, while AB is m x m and p ≠ m. Thus AB and BA are not equal
For Example : A=4x5 , B=5x4
AB and BA may both the same size, but not equal as matrices AB ≠ BA
For Example : A=4x4 , B=4x4
AB = BA
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24. Basic Properties of Multiplication
The basic properties of matrix multiplication are given by the following theorem:
A(BC) = (AB)C
A(B + C)= AB + AC
(A + B)C = AC + BC
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25. Transpose Matrices Example: The transpose of the matrix is the matrix
DEFINITION 6:
Let A = [aij] be an m x n matrix. The transpose of A, denoted by At, is the n x m matrix obtained by interchanging the rows and columns of A.
In other words, if At = [bij], then bij = aji, for i = 1,2,…,n and j = 1,2,…,m.
Example:
The transpose of the matrix
is the matrix
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26. Properties for Transpose
If A and B are matrices, then
𝐴 𝑇 𝑇 =𝐴
(𝐴+𝐵) 𝑇 = 𝐴 𝑇 + 𝐵 𝑇
(𝐴𝐵) 𝑇 = 𝐵 𝑇 𝐴 𝑇
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27. Exercises Consider the matrices : A= ,B= ,C= Find the following:
𝐶 𝑇 (𝐴+𝐵) 𝑇
=𝐴 𝑇 + 𝐵 𝑇
−4 −2
− − − =
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28. Symmetric Matrices Example: The matrix is symmetric. Example
DEFINITION 7:
A square matrix A is called symmetric if A = At.
Thus A = [aij] is symmetric if aij = aji for all i and j with 1 ≤ i ≤ n and 1 ≤ j ≤ n.
Example:
The matrix is symmetric.
Example
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29. The Transpose of a Symmetric Matrix
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30. Boolean Matrix Operation
A Boolean matrix is an m x n matrix whose entries are either zero or one.
Example:
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31. Boolean Matrix Operations (join)
Let A = [aij] and B = [bij] be m x n Boolean matrices.
We define A v B = C = [ Cij], the join of A and B, by
1 if aij =1 or bij = 1
Cij = if aij and bij are both 0
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32. Example Find the join of A and B: A = B = A v B = 1 0 1 0 1 0
1v0 0v1 1v0 0v1 1v1 0v0 =
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33. Boolean Matrix Operations (Meet)
We define A ^ B = C = [ Cij], the meet of A and B, by
if aij and bij are both 1
Cij = if aij = 0 or bij = 0
Meet & Join are the same as the addition procedure
each element with the corresponding element in the other matrix
Matrices have the same size
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34. Example Find the meet of A and B: A = B = A ^ B = 1 0 1 0 1 0
1^0 0^1 1^0 0^1 1^1 0^0 =
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35. Boolean PRODUCT
The Boolean product of A and B, denoted , 𝑨 ⊙𝑩 is the m x n Boolean matrix defined by
C𝑖𝑗= 1 , if aik = 1 and bkj = 1 for some k,1 ≤ k ≤ p 0,𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Procedure:
Select row i of A and column j of B, and arrange them side by side.
Compare corresponding entries. If even a single pair of corresponding entries consists of two 1’s, then Cij = 1, otherwise Cij = 0
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36. Example Find the Boolean product of A and B: A 3x2= B 2x3 = 𝑨 ⊙𝑩3x3=
(1 ^ 1) v (0 ^ 0) (1 ^ 1) v (0 ^ 1) (1 ^ 0) v (0 ^ 1) (0 ^ 1) v (1 ^ 0) (0 ^ 1) v (1 ^ 1) (0 ^ 0) v (1 ^ 1) (1 ^ 1) v (0 ^ 0) (1 ^ 1) v (0 ^ 1) (1 ^ 0) v (0 ^ 1) =
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37. Boolean Operations Properties
If A, B, and C are Boolean Matrices with the same sizes, then
A v B = B v A
A ^ B = B ^ A
(A v B) v C = A v (B v C)
(A ^ B) ^ C = A ^ (B ^ C)
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38. Exercises Find meet and join for A and B: Solution : Meet of A and B =
join of A and B =
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39. Exercises Find 𝑨 ⊙𝑩 Solution: 𝑨 ⊙𝑩 = 18-Jan-19
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40. Any Question Refer to chapter 3 of the book for further reading
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