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4-2
4-2 Multiplying
MultiplyingMatrices
Matrices
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
4-2 Multiplying Matrices
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
Holt Algebra 2
4-2 Multiplying Matrices
Objectives
Understand the properties of matrices
with respect to multiplication.
Multiply two matrices.
Holt Algebra 2
4-2 Multiplying Matrices
Vocabulary
matrix product
square matrix
main diagonal
multiplicative identity matrix
Holt Algebra 2
4-2 Multiplying Matrices
In Lesson 4-1, 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.
Holt Algebra 2
4-2 Multiplying Matrices
An m  n matrix A can be identified by using the
notation Am  n.
Holt Algebra 2
4-2 Multiplying Matrices
Helpful Hint
The CAR key:
Columns (of A)
As
Rows (of B)
or matrix product AB
won’t even start
Holt Algebra 2
4-2 Multiplying Matrices
Example 1A: Identifying Matrix Products
Tell whether the product is defined. If so, give
its dimensions.
A3  4 and B4  2; AB
A
34
B
42=
AB
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
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
53
P
25
The inner dimensions are not equal (3 ≠ 2), so the
matrix product is not defined.
Holt Algebra 2
4-2 Multiplying Matrices
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
45
R
43
The inner dimensions are not equal (5 ≠ 4), so the
matrix product is not defined.
Holt Algebra 2
4-2 Multiplying Matrices
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
45
Q
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 1 of W and column 1 of X as shown.
Place the result in wx11.
3(4) + –2(5)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 1 of W and column 2 of X as shown.
Place the result in wx12.
3(7) + –2(1)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 1 of W and column 3 of X as shown.
Place the result in wx13.
3(–2) + –2(–1)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 2 of W and column 1 of X as shown.
Place the result in wx21.
1(4) + 0(5)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 2 of W and column 2 of X as shown.
Place the result in wx22.
1(7) + 0(1)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 2 of W and column 3 of X as shown.
Place the result in wx23.
1(–2) + 0(–1)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 3 of W and column 1 of X as shown.
Place the result in wx31.
2(4) + –1(5)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 3 of W and column 2 of X as shown.
Place the result in wx32.
2(7) + –1(1)
Holt Algebra 2
4-2 Multiplying Matrices
Example 2A Continued
Multiply row 3 of W and column 3 of X as shown.
Place the result in wx33.
2(–2) + –1(–1)
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
Businesses can use matrix multiplication to find
total revenues, costs, and profits.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
Example 3 Continued
Fri
Sat
Sun
Video World
Star Movies
On Saturday, Video World made $851.05 and
Star Movies made $832.50.
Holt Algebra 2
4-2 Multiplying Matrices
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
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
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.
Holt Algebra 2
4-2 Multiplying Matrices
Because square matrices can be multiplied by
themselves any number of times, you can find powers
of square matrices.
Holt Algebra 2
4-2 Multiplying Matrices
Example 4A: Finding Powers of Matrices
Evaluate, if possible.
P3
Holt Algebra 2
4-2 Multiplying Matrices
Example 4A Continued
Holt Algebra 2
4-2 Multiplying Matrices
Example 4A Continued
Check Use a calculator.
Holt Algebra 2
4-2 Multiplying Matrices
Example 4B: Finding Powers of Matrices
Evaluate, if possible.
Q2
Holt Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4a
Evaluate if possible.
C2
The matrices cannot be multiplied.
Holt Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4b
Evaluate if possible.
A3
Holt Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4c
Evaluate if possible.
B3
Holt Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4d
Evaluate if possible.
I4
Holt Algebra 2
4-2 Multiplying Matrices
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3
Holt Algebra 2
4-2 Multiplying Matrices
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
not possible
4. BD
not possible
5. C3
Holt Algebra 2