Multiply matrices by following rules: matrices can be multiplied if the number of columns in the first equals the number of rows in the second. The product is an m x p matrix if the first is m x n and the second is n x p. Square matrices can be multiplied by themselves to find powers; the identity matrix I has 1s on the main diagonal and 0s elsewhere. Businesses can use matrix multiplication to calculate total revenues, costs, and profits from sales data.

1. Multiplying Matrices
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm Up
State the dimensions of each matrix.
1. [3 1 4 6] 1 4
2.
3 2
Calculate.
3. 3(–4) + (–2)(5) + 4(7) 6
4. (–3)3 + 2(5) + (–1)(12) –11

3. Objectives
Understand the properties of matrices
with respect to multiplication.
Multiply two matrices.

4. Vocabulary
matrix product
square matrix
main diagonal
multiplicative identity matrix

5. In previous lesson, you multiplied matrices by a
number called a scalar. You can also multiply matrices
together. The product of two or more matrices is the
matrix product. The following rules apply when
multiplying matrices.
• Matrices A and B can be multiplied only if the
number of columns in A equals the number of
rows in B.
• The product of an m n and an n p matrix
is an m p matrix.

6. An m n matrix A can be identified by using the
notation Am n.

7. Helpful Hint
The CAR key:
Columns (of A)
As
Rows (of B)
or matrix product AB
won’t even start

8. Example 1A: Identifying Matrix Products
Tell whether the product is defined. If so, give
its dimensions.
A3 4 and B4 2; AB
A B AB
3 4 4 2= 3 2 matrix
The inner dimensions are equal (4 = 4), so the matrix
product is defined. The dimensions of the product are
the outer numbers, 3 2.

9. Example 1B: Identifying Matrix Products
Tell whether the product is defined. If so, give
its dimensions.
C1 4 and D3 4; CD
C D
1 4 3 4
The inner dimensions are not equal (4 ≠ 3), so the
matrix product is not defined.

10. Check It Out! Example 1a
Tell whether the product is defined. If so, give
its dimensions.
P2 5 Q5 3 R4 3 S4 5
QP
Q P
5 3 2 5
The inner dimensions are not equal (3 ≠ 2), so the
matrix product is not defined.

11. Check It Out! Example 1b
Tell whether the product is defined. If so, give
its dimensions.
P2 5 Q5 3 R4 3 S4 5
SR
S R
4 5 4 3
The inner dimensions are not equal (5 ≠ 4), so the
matrix product is not defined.

12. Check It Out! Example 1c
Tell whether the product is defined. If so, give
its dimensions.
P2 5 Q5 3 R4 3 S4 5
SQ
S Q
4 5 5 3
The inner dimensions are equal (5 = 5), so the matrix
product is defined. The dimensions of the product are
the outer numbers, 4 3.

13. Just as you look across the columns of A and down
the rows of B to see if a product AB exists, you do
the same to find the entries in a matrix product.

15. Example 2A: Finding the Matrix Product
Find the product, if possible.
WX
Check the dimensions. W is 3 2 , X is 2 3.
WX is defined and is 3 3.

16. Example 2A Continued
Multiply row 1 of W and column 1 of X as shown.
Place the result in wx11.
3(4) + –2(5)

17. Example 2A Continued
Multiply row 1 of W and column 2 of X as shown.
Place the result in wx12.
3(7) + –2(1)

18. Example 2A Continued
Multiply row 1 of W and column 3 of X as shown.
Place the result in wx13.
3(–2) + –2(–1)

19. Example 2A Continued
Multiply row 2 of W and column 1 of X as shown.
Place the result in wx21.
1(4) + 0(5)

20. Example 2A Continued
Multiply row 2 of W and column 2 of X as shown.
Place the result in wx22.
1(7) + 0(1)

21. Example 2A Continued
Multiply row 2 of W and column 3 of X as shown.
Place the result in wx23.
1(–2) + 0(–1)

22. Example 2A Continued
Multiply row 3 of W and column 1 of X as shown.
Place the result in wx31.
2(4) + –1(5)

23. Example 2A Continued
Multiply row 3 of W and column 2 of X as shown.
Place the result in wx32.
2(7) + –1(1)

24. Example 2A Continued
Multiply row 3 of W and column 3 of X as shown.
Place the result in wx33.
2(–2) + –1(–1)

25. Example 2B: Finding the Matrix Product
Find each product, if possible.
XW
Check the dimensions. X is 2 3, and W is 3 2 so
the product is defined and is 2 2.

26. Example 2C: Finding the Matrix Product
Find each product, if possible.
XY
Check the dimensions. X is 2 3, and Y is 2 2.
The product is not defined. The matrices cannot be
multiplied in this order.

27. Check It Out! Example 2a
Find the product, if possible.
BC
Check the dimensions. B is 3 2, and C is 2 2 so
the product is defined and is 3 2.

28. Check It Out! Example 2b
Find the product, if possible.
CA
Check the dimensions. C is 2 2, and A is 2 3 so
the product is defined and is 2 3.

29. Businesses can use matrix multiplication to find
total revenues, costs, and profits.

30. Example 3: Inventory Application
Two stores held sales on their videos and DVDs,
with prices as shown. Use the sales data to
determine how much money each store brought
in from the sale on Saturday.
Use a product matrix to find the sales of each store
for each day.

31. Example 3 Continued
Fri Sat Sun
Video World
Star Movies
On Saturday, Video World made $851.05 and
Star Movies made $832.50.

32. Check It Out! Example 3
Change Store 2’s inventory to 6 complete and 9
super complete. Update the product matrix, and
find the profit for Store 2.
Skateboard Kit Inventory
Super
Complete
Complete
Store 1 14 10
Store 2 6 9

33. Check It Out! Example 3
Use a product matrix to find the revenue, cost, and
profit for each store.
Revenue Cost Profit
Store 1
Store 2
The profit for Store 2 was $819.

34. A square matrix is any matrix that has the same
number of rows as columns; it is an n × n matrix. The
main diagonal of a square matrix is the diagonal
from the upper left corner to the lower right corner.
The multiplicative identity matrix is any square
matrix, named with the letter I, that has all of the
entries along the main diagonal equal to 1 and all of
the other entries equal to 0.

35. Because square matrices can be multiplied by
themselves any number of times, you can find powers
of square matrices.

36. Example 4A: Finding Powers of Matrices
Evaluate, if possible.
P3

37. Example 4A Continued

38. Example 4A Continued
Check Use a calculator.

39. Example 4B: Finding Powers of Matrices
Evaluate, if possible.
Q2

40. Check It Out! Example 4a
Evaluate if possible.
C2
The matrices cannot be multiplied.

41. Check It Out! Example 4b
Evaluate if possible.
A3

42. Check It Out! Example 4c
Evaluate if possible.
B3

43. Check It Out! Example 4d
Evaluate if possible.
I4

44. Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3

45. Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2 not possible
4. BD not possible
5. C3