1. MATH 1046 Introduction to Matrices (Sections 3.1 and 3.2)
Alex Karassev

2. Matrices
An m x n matrix is a rectangular array of numbers with m rows and n columns
j-th column
i-th row

3. Some examples Coefficient matrix of a linear system Distance matrix
Graph adjacency matrix
Matrices in computer graphics
Matrices in optimization problems
Games

4. Example: distance matrix (in km)
North Bay
Toronto
Ottawa
340
360
450

5. Example: graph adjacency matrix
Timmins 1
North Bay 2
Ottawa 3 4
Toronto

6. Matrix operations Multiplication by scalars Addition and subtraction
Transpose

7. Multiplication by scalars
Example: distance matrices in km and mi
All entries are multiplied by 1/1.6 km NB T O
340
360
450 mi NB T O
340/1.6
360/1.6
450/1.6

8. Multiplication by scalars

9. Addition First Basket Apples Pears Red 3 5 Green 4 6 Second Basket1
Green 10 3
Total
Apples
Pears
Red
3+1
5+0
Green
4+10
6+3

10. Addition

11. Matrix miultiplication
Example: 2 x 2 linear system
Linear substitution

12. What system do we get in terms of y1 and y2?
Substitution:
What system do we get in terms of y1 and y2?

13. New Coefficient Matrix:

14. Matrix multiplication: 2 x 2 case

15. Matrix multiplication: 2 x 2 case
Dot product:

16. Matrix multiplication: general case
Let A = (aij) be m x n matrix and B be n x k matrix
The product AB = C = (cij) is an m x k matrix defined as follows cij = ai1 b1j + ai2 b2j+ ci3 b3j+…+ain bnj
Note: cij is the dot product of i-th row of A and j-th column of B

17. Is it possible that AB≠BA?

18. Is it possible that AB≠BA?
Yes!

19. Square matrix
A matrix is called square if m=n, i.e. the number of rows is the same as the number of columns
A square matrix has the diagonal: all entries of the form aii

20. Zero Matrix and Identity Matrix
Zero matrix: an n x n matrix such that all entries are zeros
Identity matrix: an n x n matrix such that all diagonal entries are 1, all other entries are 0

21. Properties: addition and multiplication by scalars
Addition of matrices and multiplication by scalars have the same properties as in the case of vectors or real numbers:
(A+B)+C=A+(B+C)
A+B = B+A
A+O = A
If -A = (-1)+A then A+(-A) = O
(cd)A= c(dA)
1A = A
(c+d)A = cA + dA
c(A+B) = cA +cB

22. Properties: matrix multiplication
Assuming the dimension of matrices allow to perform the operations, we have the following:
(AB)C=A(BC)
A(B+C)= AB+AC
(B+C)A= BA+CA
IA = A I = A
OA = AO = O
(cA)B= A (cB) = c(AB)

23. Powers
For a square matrix A, define Ak = A A … A (product of A with itself k times)
Question: Is it possible to have A2 = O for a non-zero A?

24. Yes!

25. Scalar matrices
Matrix A is called scalar if A = aI for some real number a
Exercise
Prove that for a scalar n x n matrix A and for any n x n matrix B we have AB= BA = aB
Prove (at least for 2 x 2 case) that in fact for an arbitrary square matrix A we have AB= BA for any B if and only if A is scalar

26. Matrix Transpose AT = B such that bij = aji
If A is m x n matrix AT is n x m matrix
If v is a row vector vT is a column vector, and vice versa
In general: columns of AT are rows of A rows of AT are columns of A
For a square matrix A this can be viewed as a flipping with respect to the diagonal:

27. Properties of transpose operation
(AT)T = ?
(cA) T= ?
(A+B)T = ?
(AB)T = ?
(Ak)T= ?

28. Properties of transpose operation
(AT)T = A
(cA) T=cAT
(A+B)T = AT+ BT
(AB)T =BTAT
(Ak)T= (AT) k
Note: in general AAT ≠ATA (exercise: find an example!); square matrices that commute with its transpose are called normal

29. Symmetric and skew-symmetric matrices
A square matrix is called
Symmetric if AT = A
Examples: distance matrix, adjacency matrix
Skew-symmetric (or antisymmetric) if AT = -A
Examples: matrices of some games

30. Symmetric and Skew-symmetric matrices
(A+AT)T= AT + A = A + AT
(AAT)T=(AT)TAT =AAT A+AT and AAT are symmetric for any square A
(A-AT)T=AT - A = -(A - AT) A-AT is skew-symmetric for any square A
Note: for any square matrix A we have A=1/2(A+AT) + 1/2 (A-AT)

31. Symmetric matrices: Exercise
Give an example of two symmetric 2x2 matrices whose product is not symmetric
Prove that the product of two symmetric square matrices A and B is symmetric if and only if AB = BA