Warm Up State the dimensions of each matrix. 1. [3 1 4 6] 2. Calculate. 3. 3(–4) + ( – 2)(5) +...
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Transcript of Warm Up State the dimensions of each matrix. 1. [3 1 4 6] 2. Calculate. 3. 3(–4) + ( – 2)(5) +...
![Page 1: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/1.jpg)
Multiplying Matrices
![Page 2: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/2.jpg)
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
![Page 3: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/3.jpg)
Understand the properties of matrices with respect to multiplication.
Multiply two matrices.
Objectives
![Page 4: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/4.jpg)
matrix productsquare matrixmain diagonalmultiplicative identity matrix
Vocabulary
![Page 5: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/5.jpg)
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.
![Page 6: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/6.jpg)
An m n matrix A can be identified by using the notation Am n.
![Page 7: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/7.jpg)
The CAR key:Columns (of A)AsRows (of B)or matrix product ABwon’t even start
Helpful Hint
![Page 8: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/8.jpg)
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.
![Page 9: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/9.jpg)
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.
![Page 10: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/10.jpg)
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
![Page 11: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/11.jpg)
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.
![Page 12: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/12.jpg)
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.
![Page 13: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/13.jpg)
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.
![Page 14: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/14.jpg)
![Page 15: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/15.jpg)
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.
![Page 16: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/16.jpg)
Example 2A Continued
Multiply row 1 of W and column 1 of X as shown. Place the result in wx11.
3(4) + –2(5)
![Page 17: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/17.jpg)
Example 2A Continued
Multiply row 1 of W and column 2 of X as shown. Place the result in wx12.
3(7) + –2(1)
![Page 18: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/18.jpg)
Example 2A Continued
Multiply row 1 of W and column 3 of X as shown. Place the result in wx13.
3(–2) + –2(–1)
![Page 19: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/19.jpg)
Example 2A Continued
Multiply row 2 of W and column 1 of X as shown. Place the result in wx21.
1(4) + 0(5)
![Page 20: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/20.jpg)
Example 2A Continued
Multiply row 2 of W and column 2 of X as shown. Place the result in wx22.
1(7) + 0(1)
![Page 21: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/21.jpg)
Example 2A Continued
Multiply row 2 of W and column 3 of X as shown. Place the result in wx23.
1(–2) + 0(–1)
![Page 22: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/22.jpg)
Example 2A Continued
Multiply row 3 of W and column 1 of X as shown. Place the result in wx31.
2(4) + –1(5)
![Page 23: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/23.jpg)
Example 2A Continued
Multiply row 3 of W and column 2 of X as shown. Place the result in wx32.
2(7) + –1(1)
![Page 24: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/24.jpg)
Example 2A Continued
Multiply row 3 of W and column 3 of X as shown. Place the result in wx33.
2(–2) + –1(–1)
![Page 25: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/25.jpg)
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.
![Page 26: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/26.jpg)
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.
![Page 27: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/27.jpg)
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.
![Page 28: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/28.jpg)
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.
![Page 29: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/29.jpg)
Businesses can use matrix multiplication to find total revenues, costs, and profits.
![Page 30: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/30.jpg)
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.
![Page 31: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/31.jpg)
Example 3 Continued
On Saturday, Video World made $851.05 and Star Movies made $832.50.
Fri Sat SunVideo World
Star Movies
![Page 32: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/32.jpg)
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
![Page 33: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/33.jpg)
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.
![Page 34: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/34.jpg)
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.
![Page 35: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/35.jpg)
Because square matrices can be multiplied by themselves any number of times, you can find powers of square matrices.
![Page 36: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/36.jpg)
Example 4A: Finding Powers of Matrices
Evaluate, if possible.
P3
![Page 37: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/37.jpg)
Example 4A Continued
![Page 38: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/38.jpg)
Example 4A Continued
Check Use a calculator.
![Page 39: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/39.jpg)
Example 4B: Finding Powers of Matrices
Evaluate, if possible.
Q2
![Page 40: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/40.jpg)
Check It Out! Example 4a
C2
Evaluate if possible.
The matrices cannot be multiplied.
![Page 41: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/41.jpg)
Check It Out! Example 4b
A3
Evaluate if possible.
![Page 42: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/42.jpg)
Check It Out! Example 4c
B3
Evaluate if possible.
![Page 43: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/43.jpg)
Check It Out! Example 4d
I4
Evaluate if possible.
![Page 44: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/44.jpg)
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3
![Page 45: Warm Up State 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.](https://reader035.fdocuments.net/reader035/viewer/2022062511/551c3adb55034693488b47ec/html5/thumbnails/45.jpg)
Lesson Quiz
Evaluate if possible.
1. AB
2. BA
3. A2
4. BD
5. C3
not possible
not possible