matrix revolutions: solving matrix equations matrix 3 mathscience innovation center betsey davis...
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Matrix Revolutions:Solving Matrix Equations
Matrix 3MathScience Innovation CenterBetsey Davis
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11101010101010101
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Matrix Revolutions B. Davis MathScience Innovation Center
It’s time to use matrices!
How would you solve 3 x = 6 for x?We need to think of multiplying, not dividing because with matrices there is no “divide”.Multiply 3x and 6 by the multiplicative inverse of 3.
(1/3)3x = (1/3)6 so…. X = 2
Matrix Revolutions B. Davis MathScience Innovation Center
The Big Idea
If [A] [x] = [B] where A, B, x are matrices,
Then [A]-1[A] [x] = [A]-1[B]
So [x] = [A]-1[B]
Matrix Revolutions B. Davis MathScience Innovation Center
(1/3)3x = (1/3)6
Multiplication by real numbers is commutative, so order is not important.Multiplication by matrices is NOT commutative, so order is VERY important.Let’s solve for x:
43
21X =
24
17
Matrix Revolutions B. Davis MathScience Innovation Center
First, find the inverse of the left matrix
43
21X =
24
17
2
1
2
312
2
1
2
32
2
2
4
13
24
64
1
43
211
Matrix Revolutions B. Davis MathScience Innovation Center
Second, multiply both sides of the equation by A-1
43
21X =
24
17
24
17
2
1
2
312
43
21
2
1
2
312
x
Matrix Revolutions B. Davis MathScience Innovation Center
Third, simplify both sides.
43
21X =
24
17
24
17
2
1
2
312
43
21
2
1
2
312
x
24
17
2
1
2
312
10
01x
24
17
2
1
2
312
x
Matrix Revolutions B. Davis MathScience Innovation Center
To simplify the right hand side, multiply the 2 matrices.
43
21X =
24
17
24
17
2
1
2
312
x
2
5
2
17410
2*2
11*
2
34*
2
17*
2
32*11*24*17*2
x
Matrix Revolutions B. Davis MathScience Innovation Center
By Calculator:
43
21X =
24
17
24
17*
43
211
x
So, just enter A and B into the calculator.
Then on the home screen type [A] x-1 [B] enter.
Matrix Revolutions B. Davis MathScience Innovation Center
43
21X =
24
17
Final Answer for x.
2
5
2
17410
x
Matrix Revolutions B. Davis MathScience Innovation Center
SO WHAT???????
We can use this new skill– solving equations using matrices –
To solve linear
systems.
Matrix Revolutions B. Davis MathScience Innovation Center
Let’s learn how!
Basic idea comes from solving AX = BLet’s write a system:
3 x + 2 y = 6 5 x - 9 y = 15Now, let’s rewrite the system using matrices:
Matrix Revolutions B. Davis MathScience Innovation Center
3 x + 2 y = 6 5 x - 9 y = 15
Re-writing the system using matrices:
•Make a matrix of coefficients.
•Make a matrix of variables.
•Make a matrix of constants.
95
23
y
x
15
6=
Matrix Revolutions B. Davis MathScience Innovation Center
3 x + 2 y = 6 5 x - 9 y = 15
What size are these and can they be multiplied?
What size is the answer?
95
23
y
x
15
6=
2 x 2 2 x 1 2 x 1
Matrix Revolutions B. Davis MathScience Innovation Center
3 x + 2 y = 6 5 x - 9 y = 15
• We know we can multiply them and the answer is a 2x1.
• We will use the same BIG IDEA:
If [A][x]=[B], then [x] = [A] –1 [B]
95
23
y
x
15
6=
Matrix Revolutions B. Davis MathScience Innovation Center
Here we go!
95
23
y
x
15
6=
Remember BIG IDEA:
If [A][x]=[B], then [x] = [A] –1 [B]
1
95
23
1
95
23
Important!!! Notice order of multiplication
Matrix Revolutions B. Davis MathScience Innovation Center
We have identity matrix on left
y
x
15
6=
1
95
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10
01
Matrix Revolutions B. Davis MathScience Innovation Center
The identity times [x] is [x].
y
x
15
6=
1
95
23
Now just type [A]-1[B] on your TI
Matrix Revolutions B. Davis MathScience Innovation Center
y
x=
37
1537
84
What does this mean?
The solution is the ordered pair
37
15,
37
84
Final answer
Matrix Revolutions B. Davis MathScience Innovation Center
Let’s try it !
2w – x + 5 y + z = -33w + 2x + 2 y – 6 z = -32 w + 3x + 3 y - z = -475w – 2 x - 3 y + 3 z = 49
We will need 3 matrices…
Matrix Revolutions B. Davis MathScience Innovation Center
Matrix of Coefficients
2w – 1 x + 5 y + 1z = -33w + 2x + 2 y – 6 z = -321w + 3x + 3 y -1 z = -475w – 2 x - 3 y + 3 z = 49
3325
1331
6223
1512
Matrix Revolutions B. Davis MathScience Innovation Center
Matrix of variables
2w – x + 5 y + z = -33w + 2x + 2 y – 6 z = -32 w + 3x + 3 y - z = -475w – 2 x - 3 y + 3 z = 49
z
y
x
w
Matrix Revolutions B. Davis MathScience Innovation Center
Matrix of constants
2w – x + 5 y + z = -33w + 2x + 2 y – 6 z = -32 w + 3x + 3 y - z = -475w – 2 x - 3 y + 3 z = 49
49
47
32
3
Matrix Revolutions B. Davis MathScience Innovation Center
49
47
32
3
z
y
x
w
=
3325
1331
6223
1512
Matrix Revolutions B. Davis MathScience Innovation Center
49
47
32
3
z
y
x
w
=
3325
1331
6223
1512
Matrix Revolutions B. Davis MathScience Innovation Center
49
47
32
3
z
y
x
w
=
3325
1331
6223
1512
Matrix Revolutions B. Davis MathScience Innovation Center
49
47
32
3
z
y
x
w
=
3325
1331
6223
1512
Matrix Revolutions B. Davis MathScience Innovation Center
49
47
32
3
z
y
x
w
=
3325
1331
6223
1512-1
Matrix Revolutions B. Davis MathScience Innovation Center
1
4
12
2
z
y
x
w
=
Therefore,
The solution is
An ordered quadruplet:
(2,-12,-4, 1)
Matrix Revolutions B. Davis MathScience Innovation Center
Way cool, huh?