1. Math-2 (honors) Matrix Algebra
Lesson 11.4

2. What you’ll learn Matrices Matrix Addition and Subtraction
Matrix Multiplication
Identity and Inverse Matrices
Determinant of a Square Matrix
Applications
… and why
Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

3. Vocabulary Matrix: A rectangular arrangement of
numbers in rows and columns.
Dimensions (order): Of a matrix with 3 rows and 2 columns is: 3 x 2
In general we say: m x n where:
“m” = # of rows
“n” = # of Columns

4. What is a Matrix?
A rectangular arrangement of numbers in rows and columns
What is the order or dimensions of this matrix?

5. Matrix Vocabulary
Each element, or entry, aij, of the matrix uses double subscript notation.
Row subscript: is the 1st letter (i)
Column Subscript: is the 2nd letter (j)
Example: Element aij is in the ith row and jth column.
Is “shorthand” for a matrix with ‘i’ rows and
‘j’ columns. (‘i’ x ‘j’)

6. Vocabulary 5 -2 3 1 Elements: numbers in the matrix What is the
“order” of this
matrix ?
1,1
1,2 3 1
2,2
2,1
Equal matrices: have same “order” and each corresponding element is equal.
Your turn: What number is
2. What number is

7. Matrices can be HUGE ! 5 columns 4 rows Dimension: m rows x n columns
4 rows
Dimension: m rows x n columns
4 x 5

8. Vocabulary
Scalar: A real number (a constant) that is multiplied by every element in the matrix.
Scalar Multiplication: The process of multiplying every element in the matrix by a scalar (constant).
Let: ‘A’ represent the matrix
Then: 3A = 3

9. Multiplying by a constant (also called a “Scalar Multiplication”)
3 4
9 1
7 -2
5(3) 5(4)
5(9) 5(1)
5(7) 5(-2) 5 = =

10. Your turn:
3. Write the result of the following:
2 –3 -4

11. Basic Operations: Addition
2 –3 ? + =
A B
Matrices are added “corresponding element” to “corresponding element”

12. Basic Operations: Addition
2 –3
(2+3) + =

13. Basic Operations: Addition
2 –3 5 + =

14. Basic Operations: Addition
2 –3 5
(–3+1) + =

15. Basic Operations: Addition
2 –3 5 -2 + =

16. Basic Operations: Addition
2 –3 5 -2 + =
(7-3)

17. Basic Operations: Addition
2 –3 5 -2 + = 4

18. Basic Operations: Addition & Subtraction
2 –3 5 -2 + = 4
(2+5)

19. Basic Operations: Addition & Subtraction
2 –3 5 -2 + = 4 7

20. Basic Operations: Addition
2 –3 ? + =
CAN’T DO THIS!!!!!!
(must be the same order for addition/subtraction)

21. Your turn:
4. Write the result of the following:
2 –3 2 +

22. m x n (times) n x p = m x p m x p
Order of the “Product” of Two Matrices (2 matrices multiplied by each other):
Matrix A x Matrix B = AB
m x n (times) n x p
= m x p
m x p
Must be equal !!

23. Matrix Multiplication
A x B = ?
2 –3 ? x =
2 x x 2
2 x 2 =
equal
m x p

24. What is the dimension of the product?
2 –3 x ? =
2 x x 3
2 x 3 =
equal
m x p

25. What is the dimension of the product?
2 –3 x ? =
2 x x 2
NOTequal !!!!!!!
CAN’T DO THIS!!!!!!

26. So, how do you multiply matrices?
1 3
4 2 x =
1,1
1,2
2,1
2,2
Can you multiply?
What is order of
the answer matrix?

27. So, how do you multiply matrices?
1 3
4 2 x =
(1*2+3*1)
1,1
1,2
2,1
2,2
What is the “address”
Of this element?
1,1: (1st row, 1st column)

28. So, how do you multiply matrices?
1 3
4 2 x = 5
1,1
1,2
2,1
2,2

29. So, how do you multiply matrices?
1 3
4 2 x = 5
(1*-1+3*-2)
1,1
1,2
2,1
2,2

30. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
2,1
2,2

31. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
(4*2+2*1)
2,1
2,2

32. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2 10
2,1
2,2

33. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2
(4*-1+2*-2) 10
2,1
2,2

34. So, how do you multiply matrices?
1 3
4 2 x = 5 -7
1,1
1,2 -8 10
2,1
2,2

35. Your turn:
5. Write the product of the
two matrices.
2 –3 x ? =
2 x x 3
2 x 3 =
equal
m x p

36. Is Matrix Multiplication commutative?
B = AB -7
1 3
4 2 = x
Your Turn:
1 3
4 2 x
= ?
7. Write the product of the
two matrices.
8. Does AB = BA ?
B * A = AB

37. Matrix Multiplication using the Ti-84
2nd Matrix
Edit, Enter order
Enter elements
2nd matrix
Edit (scroll to “B” matrix
and enter elements
Clear screen (2nd “quit”)
2nd matrix, scroll to desired
matrix and enter
Enter operation desired “*”
2nd matrix, scroll to other matrix and enter, then enter again.