an m n matrix a can be identified by using the notation a m n
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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. - PowerPoint PPT PresentationTRANSCRIPT
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
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 Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3
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 Algebra 2
4-2 Multiplying Matrices
Tell whether the product is defined. If so, give its dimensions.
P2 5 Q5 3
P Q
2 5 5 3
Check It Out! Example 1b
PQ
The inner dimensions are equal (5 = 5), so the matrix product will be a 2 x 3 matrix.
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
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
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 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 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 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