1. Lesson 12 – 2 Multiplication of Matrices
Pre-calculus

2. Learning Objective
To solve quadratic systems

3. Matrix Matrix Multiplication
The product of two matrices, 𝐴 𝑚 𝑥 𝑝 and 𝐵 𝑝 𝑥 𝑛 , is the matrix 𝐴𝐵 with dimensions 𝑚 𝑥 𝑛. Any element in the 𝑖 𝑡ℎ row and 𝑗 𝑡ℎ column of this product matrix is the sum of the products of the corresponding elements of the 𝑖 𝑡ℎ row of 𝐴 and the 𝑗 𝑡ℎ column of 𝐵
Don’t Write. You have it!
When you multiply matrices, they need to be conformable, for multiplication.
Yes, write it
Extra 
This means: # of columns in 1st matrix = # of rows in 2nd matrix

4. Matrix 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 2 x 3 3 x 2 MUST MATCH TO MULTIPLY
Multiplying Matrices – order MATTERS
Matrix
𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓
2 x 3
3 x 2
MUST MATCH TO MULTIPLY
Answer will be a 2 x 2

5. Matrix 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑 3 x 2 2 x 2 MUST MATCH TO MULTIPLY
𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑
3 x 2
2 x 2
MUST MATCH TO MULTIPLY
Answer will be a 3 x 2

6. Matrix 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 2 x 3 2 x 3
𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 ∙ 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓
2 x 3
2 x 3
Don’t match, CANNOT MULTIPLY

7. Matrix 1. Find 𝐴𝐵: 𝐴= 4 5 7 2 1 3 and 𝐵= 2 3 5 6 3 x 2 2 x 2 match
dimensions of product
3 × 2
This is the first row, first column so we take 1st row of A × 1st column of B
write yourself how to get this element
𝑎 𝑑 𝑏 𝑒 𝑐 𝑓
4 2 +5(5)
4 3 +5(6)
7 2 +2(5)
7 3 +2(6)
1 2 +3(5)
1 3 +3(6)
𝐴𝐵=

8. 2. Find 𝐴 2 : 𝐴=
Matrix
𝐴 2 =
=𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
3 x 2
3 x 2
Wait! You Can’t!

9. Matrix We can solve for unknown elements in a matrix equation
3. Solve for x & y
3 1 − 𝑥 4 −4 5𝑦 = 2 −3 −4 −8
3x – 4 = 2
12 + 5y = –3
x = 2
y = –3

10. Matrix The Identity Matrix
The identity matrix is the equivalent to the algebraic 1.
Multiplying by it does not change the original.
Yes, write it 
𝐼 3 𝑥 3 =
𝐼 2 𝑥 2 =
Pattern: 1’s along the diagonal & 0’s everywhere else
*If the product of two matrices is 𝐼, then they are inverses of each other.
You can also multiply by a 0 matrix to get a 0 matrix.

11. Matrix Properties of Matrix Multiplication for Square Matrices
If 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are 𝑛 𝑥 𝑛 matrices, then
𝐴𝐵 is an 𝑛 𝑥 𝑛 matrix
Closure
Don’t Write. You have it!
𝐴𝐵 𝐶=𝐴(𝐵𝐶)
Associative
𝐼 𝑛 𝑥 𝑛 𝐴=𝐴 𝐼 𝑛 𝑥 𝑛 =𝐴
Multiplicative Identity
𝑂 𝑛 𝑥 𝑛 𝐴=𝐴 𝑂 𝑛 𝑥 𝑛 = 𝑂 𝑛 𝑥 𝑛
Multiplicative Property of the Zero Matrix
𝐴 −1 is the multiplicative inverse of 𝐴 if 𝐴 −1 is defined and 𝐴 𝐴 −1 = 𝐴 −1 𝐴= 𝐼 𝑛 𝑥 𝑛
Multiplicative Inverse

12. Matrix Properties of Matrix Multiplication for Square Matrices
If 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 are 𝑛 𝑥 𝑛 matrices, then
Don’t Write. You have it!
𝐴 𝐵+𝐶 =𝐴𝐵+𝐴𝐶
Distributive Properties
𝐵+𝐶 𝐴=𝐵𝐴+𝐶𝐴
What properties are not here??
Yes, write it
Extra 
When we “store” information in matrices, we may have to transpose them (switch rows & columns) to make them conformable for multiplication.
Per 3
Per 4
Boys
Girls
Ex.
𝐴= 𝐵𝑜𝑦𝑠 𝐺𝑖𝑟𝑙𝑠
𝐴′= 𝑃𝑒𝑟 3 𝑃𝑒𝑟
It’s still the same!

13. Matrix Don’t Write. You have it! Cost Yes, write it 
4. A fruit stand owner packages fruit in three different ways for gift packages. Economy package, E, has 6 apples, 3 oranges, and 3 pears. Standard package, S has 5 apples, 4 oranges, and 4 pears. Luxury package, L, has 6 types of each fruit. The costs are $0.50 for an apple, $1.10 for an orange, and $0.80 for a pear. What is the total cost of preparing each package of fruit?
Matrix
Don’t Write. You have it!
Cost
Apple Orange Pear
𝐴=𝑐𝑜𝑠𝑡 [ $0.50 $1.10 $0.80 ]
Yes, write it 
Number of Items
Apple Orange Pear
𝐵= 𝐸 𝑆 𝐿

14. Matrix Cost Apple Orange Pear 𝐴=𝑐𝑜𝑠𝑡 [ $0.50 $1.10 $0.80 ]
Number of Items
Apple Orange Pear
𝐵= 𝐸 𝑆 𝐿
If we multiply in this state … the labels don’t match up
Cost per fruit package per fruit
Cost per fruit fruit per package

15. 𝐴 𝐵 𝑡 = $0.50 $1.10 $
Matrix
E S L
=cost $8.70 $10.10 $14.40

16. Assignment
Pg. 608 #1, 3, 5, 8, 15, 16, 18, 19, 20, 29, 31, 34, 41, 42

17. Lesson 12 – 2 Multiplication of Matrices
Pre-calculuS
SUPPLEMENTAL NOTES

18. Multiplying Matrices 5. 𝐴= 2 8 −3 0 and 𝐵= −2 −8 3 0 find 𝐴𝐵
2 8 − −2 −8 3 0
= 2 −2 +8(3) 2 −8 +8(0) −3 −2 +0(3) −3 −8 +0(0)
= 2 −2 +8(3) 2 −8 +8(0) −3 −2 +0(3)
= 2 −2 +8(3) 2 −8 +8(0)
= 2 −2 +8(3)
= 20 −

19. Multiplying Matrices 6. 𝐴= 2 −1 3 4 and 𝐵= −3 1 0 2 find 𝐴𝐵
2 − −
= 2 −3 +(−1)(0) 2 1 +(−1)(2) 3 −3 +4(0) (2)
= 2 −3 +(−1)(0) 2 1 +(−1)(2) 3 −3 +4(0)
= 2 −3 +(−1)(0) 2 1 +(−1)(2)
= 2 −3 +(−1)(0)
= −6 0 −9 11

20. Multiplying Matrices 7. 𝐴= 2 −1 3 4 and 𝐵= −3 1 0 2 find 𝐵𝐴
− −1 3 4
= −3 2+(1)(3) −3 (−1)+(1)(4) (3) 0 −1 +2(4)
= −3 2+(1)(3) −3 (−1)+(1)(4) (3)
= −3 2+(1)(3)
= −3 2+(1)(3) −3 (−1)+(1)(4)
= −
***NOT the same as AB (#6)!!!!

21. Multiplying Matrices 8. −1 1 1 4 −3 5 (1 x 2)(2 x 2) = 1 x 2
8. − −3 5
(1 x 2)(2 x 2)
= 1 x 2
= − (−3) − (5)
= − (−3)
= −4 1

22. Multiplying Matrices
− −1 1
(2 x 2)(1 x 2)
Can’t multiply!

23. Multiplying Matrices 10. −1 −2 0 2 3 1 −1 2 −3 2 0 1 (3 x 2)(2 x 3)
10. −1 − −1 2 −
(3 x 2)(2 x 3)
= 3 x 3
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2) (0) 0 −3 +2(1)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2) (0) 0 −3 +2(1) 3 −1 +1(2)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2) (0)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2) (0) 0 −3 +2(1) 3 −1 +1(2) (0) 3 −3 +1(1)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2) (0) 0 −3 +2(1) 3 −1 +1(2) (0)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1)
= −1 −1 +(−2)(2) −1 2 +(−2)(0) −1 −3 +(−2)(1) 0 −1 +2(2)
= −1 −1 +(−2)(2) −1 2 +(−2)(0)
= −1 −1 +(−2)(2)
= −3 − −1 6 −8

24. Multiplying Matrices 11. −1 2 −3 2 0 1 −1 −2 0 2 3 1 (2 x 3)(3 x 2)
−1 2 − −1 −
(2 x 3)(3 x 2)
= 2 x 2
= −1 − (−3)(3)
= −1 − (−3)(3) −1 − (−3)(1)
= −1 − (−3)(3) −1 − (−3)(1) 2 − (3)
= −1 − (−3)(3) −1 − (−3)(1) 2 − (3) 2 − (1)
= −8 3 1 −3