4.3 matrix multiplication 1.multiplying a matrix by a scalar 2.multiplying matrices
TRANSCRIPT
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4.3 Matrix Multiplication
1. Multiplying a Matrix by a Scalar
2. Multiplying Matrices
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1) Multiplying a Matrix by a Scalar
The number or factor you multiply a matrix by is called a scalar.
When multiplying by a scalar, every element in the matrix gets multiplied.
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1) Multiplying a Matrix by a Scalar
Example 1:
Find 3-2 -3 4
0 9 3.4
3 is a scalar
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1) Multiplying a Matrix by a Scalar
Example 1:
Find 3
=
-2 -3 4
0 9 3.4
-6 -9 12
0 27 10.2
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1) Multiplying a Matrix by a Scalar
Example 2: Find X.
6 9
-12 15
27 -18
30 6=-3X + 2
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1) Multiplying a Matrix by a Scalar
Example 2: Find X.
6 9
-12 15
27 -18
30 6=-3X + 2
12 18
-24 30
27 -18
30 6=-3X +
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1) Multiplying a Matrix by a Scalar
Example 2: Find X.
6 9
-12 15
27 -18
30 6=-3X + 2
12 18
-24 30
27 -18
30 6=-3X +
12 18
-24 30
27 -18
30 6
=-3X -
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1) Multiplying a Matrix by a Scalar
Example 2: Find X.
6 9
-12 15
27 -18
30 6=-3X + 2
12 18
-24 30
27 -18
30 6=-3X +
15 -36
54 -24
=-3X
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1) Multiplying a Matrix by a Scalar
Example 2: Find X.
6 9
-12 15
27 -18
30 6=-3X + 2
12 18
-24 30
27 -18
30 6=-3X +
-5 12
-18 8
=X
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2) Multiplying Matrices
Multiply the rows of the first matrix by the columns of the second matrix and add.
Stay organized or it will get messy…fast.
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2) Multiplying Matrices
Example 1:
Find the product of and-3 3
5 0
-1 0
3 -4
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2) Multiplying Matrices
Example 1:
Find the product of and
Answer: Some matrix with dimensions 2x2
-3 3
5 0
-1 0
3 -4
? ?
? ?
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2) Multiplying Matrices
Example 1:
Find the product of and-3 3
5 0
-1 0
3 -4
Multiply row 1 and column 1
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2) Multiplying Matrices
Example 1:
Find the product of and
= (-3)(-1) + 3(3)
= 12
-3 3
5 0
-1 0
3 -4
Multiply row 1 and column 1
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2) Multiplying Matrices
Example 1:
Find the product of and
= (-3)(-1) + 3(3)
= 12
-3 3
5 0
-1 0
3 -4
Multiply row 1 and column 1
12 ?
? ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (-3)(0) + 3(-4)
= -12
-3 3
5 0
-1 0
3 -4
Multiply row 1 and column 2
12 ?
? ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (-3)(0) + 3(-4)
= -12
-3 3
5 0
-1 0
3 -4
Multiply row 1 and column 2
12 -12
? ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (5)(-1) + 0(3)
= -5
-3 3
5 0
-1 0
3 -4
Multiply row 2 and column 1
12 -12
? ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (5)(-1) + 0(3)
= -5
-3 3
5 0
-1 0
3 -4
Multiply row 2 and column 1
12 -12
-5 ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (5)(0) + 0(-4)
= 0
-3 3
5 0
-1 0
3 -4
Multiply row 2 and column 2
12 -12
-5 ?
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2) Multiplying Matrices
Example 1:
Find the product of and
= (5)(0) + 0(-4)
= 0
-3 3
5 0
-1 0
3 -4
Multiply row 2 and column 2
12 -12
-5 0
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2) Multiplying Matrices
Example 2:
Find the product of and-3 3
5 0
-1 0
3 -4
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2) Multiplying Matrices
Example 2:
Find the product of and-3 3
5 0
-1 0
3 -4
? ?
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and-3 3
5 0
-1 0
3 -4
? ?
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (-1)(-3) + (0)(5)
= 3
-3 3
5 0
-1 0
3 -4
? ?
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (-1)(-3) + (0)(5)
= 3
-3 3
5 0
-1 0
3 -4
3 ?
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (-1)(3) + (0)(5)
= -3
-3 3
5 0
-1 0
3 -4
3 ?
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (-1)(3) + (0)(5)
= -3
-3 3
5 0
-1 0
3 -4
3 -3
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (3)(-3) + (-4)(5)
= -29
-3 3
5 0
-1 0
3 -4
3 -3
? ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (3)(-3) + (-4)(5)
= -29
-3 3
5 0
-1 0
3 -4
3 -3
-29 ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (3)(3) + (-4)(0)
= 9
-3 3
5 0
-1 0
3 -4
3 -3
-29 ?
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2) Multiplying Matrices
Example 2:
Find the product of and
= (3)(3) + (-4)(0)
= 9
-3 3
5 0
-1 0
3 -4
3 -3
-29 9
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4.2 Quiz Prep
p.180 #35-38
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4.2 Quiz Prep Answers
p.180 #35-38
35) B
36) G
37) x = 4, y = -2, w = 0, z = 6
38) x = 5, y = 1
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
**The matrices are different dimensions
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
**The matrices are different dimensions
**How do you know if a matrix product exists?
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B = -2 5 2
5 0.5 1
-3 7
4 2
AB BA
(2 x 2)(2 x 3) (2 x 3)(2 x 2)
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2) Multiplying Matrices
Example 3:
Find the product of A = and B = -2 5 2
5 0.5 1
-3 7
4 2
AB BA
(2 x 2)(2 x 3) (2 x 3)(2 x 2)
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2) Multiplying Matrices
Example 3:
Find the product of A = and B = -2 5 2
5 0.5 1
-3 7
4 2
AB BA
(2 x 2)(2 x 3) (2 x 3)(2 x 2)
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2) Multiplying Matrices
Example 3:
Find the product of A = and B = -2 5 2
5 0.5 1
-3 7
4 2
AB BA
(2 x 2)(2 x 3) (2 x 3)(2 x 2)
Columns of A match rows of B
Columns of B do not match rows of A
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2) Multiplying Matrices
Example 3:
Find the product of A = and B = -2 5 2
5 0.5 1
-3 7
4 2
AB BA
(2 x 2)(2 x 3) (2 x 3)(2 x 2)
Columns of A match rows of B
Columns of B do not match rows of A
Product AB is defined
Product BA is undefined
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
Dimensions of the product matrix
= (2 x 2)(2 x 3)
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
Dimensions of the product matrix
= (2 x 2)(2 x 3)
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
Dimensions of the product matrix
= (2 x 2)(2 x 3)
= (2 x 3)
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
-3(-2) + 7(5) -3(5) + 7(0.5) -3(2) + 7(1)
=
4(-2) + 2(5) 4(5) + 2(0.5) 4(2) + 2(1)
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 3:
Find the product of A = and B =
41 -11.5 1
=
2 21 10
-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 4:
Find the product of B = and A =-2 5 2
5 0.5 1
-3 7
4 2
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2) Multiplying Matrices
Example 4:
Find the product of B = and A =
= (2 x 3)(2 x 2)
= undefined
-2 5 2
5 0.5 1
-3 7
4 2
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Homework
p.186 #5, 7, 10, 13-15, 19, 20-24, 40, 51
Tomorrow
In-class assignment
Friday
Quiz – section 4.3
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Select Homework Solutions
p.186 #5, 7, 10, 13-15, 19, 20-24, 40, 51
14) [34] 23) Undefined15) [34 0] 24) Defined20) Defined 51) x = -3, y = -921) Defined22) Defined