1. Lesson 54 – Multiplication of Matrices
Math 2 Honors - Santowski
IB Math SL - Santowski

2. Lesson Objectives
(1) Review simple terminology associated with matrices
(2) Review simple operations with matrices (+,-, scalar multiplication)
(3) Compare properties of numbers with matrices (and at the same time introduce the use of the GDC)
(4) Multiply matrices
IB Math SL - Santowski

3. (F) Properties of Matrix Addition
Ordinary Algebra with Real numbers
Matrix Algebra
if a and b are real numbers, so is the product a+b
Commutative:
a + b = b + a for all a,b
Associative:
(a + b) + c = a + (b + c)
Additive Identity:
a + 0 = 0 + a = a for all a
Additive Inverse:
a + (-a) = (-a) + a = 0
IB Math SL - Santowski

4. TI-84 and Matrices
Here are the screen captures on HOW to use the TI-84 wherein we test our properties of matrix addition
Use 2nd x-1 to access the matrix menu
IB Math SL - Santowski

5. (F) Properties of Matrix Addition
Ordinary Algebra with Real numbers
Matrix Algebra
if a and b are real numbers, so is the product a+b
if A and B are matrices, so is the sum A + B provided that …..
Commutative:
a + b = b + a for all a,b
in general, A + B = B + A provided that …..
Associative:
(a + b) + c = a + (b + c)
(A + B) + C = A + (B + C) is true provided that …..
Additive Identity:
a + 0 = 0 + a = a for all a
A + 0 = 0 + A = A for all A where 0 is the zero matrix
Additive Inverse:
a + (-a) = (-a) + a = 0
A + (-A) = (-A) + A = 0
IB Math SL - Santowski

6. Multiplying Matrices - Generalized Example
If we multiply a 2×3 matrix with a 3×1 matrix, the product matrix is 2×1
Here is how we get M11 and M22 in the product.
M11 = r11× t11  +  r12× t21  +   r13×t31
M12 = r21× t11  +  r22× t21   +  r23×t31
IB Math SL - Santowski

7. (B) Matrix Multiplication - Summary
Summary of Multiplication process
IB Math SL - Santowski

8. (D) Examples for Practice
Multiply the following matrices:
IB Math SL - Santowski

9. (D) Examples for Practice
Multiply the following matrices:
IB Math SL - Santowski

10. (E) Examples for Practice – TI-84
Here are the key steps involved in using the TI-84
IB Math SL - Santowski

11. (E) Examples for Practice – TI-84
Here are the key steps involved in using the TI-84
IB Math SL - Santowski

12. (F) Properties of Matrix Multiplication
Now we pass from the concrete to the abstract  What properties are true of matrix multiplication where we simply have a matrix (wherein we know or don’t know what elements are within)
Asked in an alternative sense  what are the general properties of multiplication (say of real numbers) in the first place???
IB Math SL - Santowski

13. (F) Properties of Matrix Multiplication
Ordinary Algebra with Real numbers
Matrix Algebra
if a and b are real numbers, so is the product ab
ab = ba for all a,b
a0 = 0a = 0 for all a
a(b + c) = ab + ac
a x 1 = 1 x a = a
an exists for all a > 0
IB Math SL - Santowski

14. (C) Key Terms for Matrices
We learned in the last lesson that there is a matrix version of the addition property of zero.
There is also a matrix version of the multiplication property of one.
The real number version tells us that if a is a real number, then a*1 = 1*a = a.
The matrix version of this property states that if A is a square matrix, then A*I = I*A = A, where I is the identity matrix of the same dimensions as A.
Definition  An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere.
IB Math SL - Santowski

15. (C) Key Terms for Matrices
Definition  An identity matrix is a square matrix with ones along the main diagonal and zeros elsewhere.
So, in matrix multiplication  A x I = I x A = A
IB Math SL - Santowski

16. (F) Properties of Matrix Multiplication
Ordinary Algebra with Real numbers
Matrix Algebra
if a and b are real numbers, so is the product ab
if A and B are matrices, so is the product AB
ab = ba for all a,b
in general, AB ≠ BA
a0 = 0a = 0 for all a
A0 = 0A = 0 for all A where 0 is the zero matrix
a(b + c) = ab + ac
A(B + C) = AB + AC
a x 1 = 1 x a = a
AI = IA = A where I is called an identity matrix and A is a square matrix
an exists for all a > 0
An for {n E I | n > 2} and A is a square matrix
IB Math SL - Santowski

17. (F) Properties of Matrix Multiplication
This is a good place to use your calculator if it handles matrices. Do enough examples of each to convince yourself of your answer to each question
(1) Does AB = BA for all B for which matrix multiplication is defined if ?
(2) In general, does AB = BA?
(3) Does A(BC) = (AB)C?
(4) Does A(B + C) = AB + AC?
IB Math SL - Santowski

18. (F) Properties of Matrix Multiplication
This is a good place to use your calculator if it handles matrices. Do enough examples of each to convince yourself of your answer to each question
(6) Does A - B = -(B - A)?
(7) For real numbers, if ab = 0, we know that either a or b must be zero. Is it true that AB = 0 implies that A or B is a zero matrix?
IB Math SL - Santowski

19. Internet Links
IB Math SL - Santowski