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Holt McDougal Algebra 2
4-2 Multiplying Matrices4-2 Multiplying Matrices
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Warm UpState the dimensions of each matrix.
1. [3 1 4 6]
2.
Calculate.
3. 3(–4) + (–2)(5) + 4(7)
4. (–3)3 + 2(5) + (–1)(12)
1 4
3 2
6
–11
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Understand the properties of matrices with respect to multiplication.
Multiply two matrices.
Objectives
Holt McDougal Algebra 2
4-2 Multiplying Matrices
matrix productsquare matrixmain diagonalmultiplicative identity matrix
Vocabulary
Holt McDougal 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 McDougal Algebra 2
4-2 Multiplying Matrices
An m n matrix A can be identified by using the notation Am n.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
The CAR key:Columns (of A)AsRows (of B)or matrix product ABwon’t even start
Helpful Hint
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
Example 1A: Identifying Matrix Products
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.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
Example 1B: Identifying Matrix Products
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 McDougal Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
Q P
5 3 2 5
The inner dimensions are not equal (3 ≠ 2), so the matrix product is not defined.
Check It Out! Example 1a
QP
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
S R
4 5 4 3
Check It Out! Example 1b
SR
The inner dimensions are not equal (5 ≠ 4), so the matrix product is not defined.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3 R4 3 S4 5
S Q
4 5 5 3
Check It Out! Example 1c
SQ
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 McDougal 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 McDougal Algebra 2
4-2 Multiplying Matrices
Holt McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal Algebra 2
4-2 Multiplying Matrices
Businesses can use matrix multiplication to find total revenues, costs, and profits.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
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.
Example 3: Inventory Application
Use a product matrix to find the sales of each store for each day.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Example 3 Continued
On Saturday, Video World made $851.05 and Star Movies made $832.50.
Fri Sat SunVideo World
Star Movies
Holt McDougal 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
CompleteSuper
Complete
Store 1 14 10
Store 2 6 9
Holt McDougal 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 ProfitStore 1Store 2
The profit for Store 2 was $819.
Holt McDougal 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 McDougal 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 McDougal Algebra 2
4-2 Multiplying Matrices
Example 4A: Finding Powers of Matrices
Evaluate, if possible.
P3
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Example 4A Continued
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Example 4A Continued
Check Use a calculator.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Example 4B: Finding Powers of Matrices
Evaluate, if possible.
Q2
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4a
C2
Evaluate if possible.
The matrices cannot be multiplied.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4b
A3
Evaluate if possible.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4c
B3
Evaluate if possible.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Check It Out! Example 4d
I4
Evaluate if possible.
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3
Holt McDougal Algebra 2
4-2 Multiplying Matrices
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3
not possible
not possible