Tuesday, August 31, 2010

MATRIX MADNESS

First off, don't mess this up:
Singular: matrix
Plural: matrices

Fun facts about matrices:)
A Matrix is a rectangular array of real numbers called entries (spreadsheet)



They have dimensions!
rows x columns (order matters)

Ex: Write dimensions of this matrix.



ANS: 2x3

You can add and subtract them!
But be careful kids... they must have the same dimensions

Ex:


ANS: They're incompatible. Gotcha!

Ex:
!


You can even multiply them together!
-AxB is possible if and only if the number of columns in A equals the number of rows in B


Ex: If A(2x3) and B(3x1), the answer will be (2x1)


Who are you matrix? Show us your identity


Matrix identities are defined, but only for square matrices (nxn)


for a 2x2:





for a 3x3:



Determinants!
ewww... what's that?

Definition:A DETERMINANT is any real number associated with any square matrix
-vertical bars on both sides of th! e matrix can also denote a matrix

Determinant of a 2x2 Matrix:

Coefficient Matrix Determinant







determinant (A) = IAI= ad-bc


NOTE: The determinant is the difference of the products of the 2 diagonals of the matrix

Ex: Find the! determinant of the matrix


Ans: det(A)= 2(2)- 1(-3) = 4+3=7


USING A CALCULATOR!
Knowing this will make your life ten times easier, maybe even 100...
1. Hit 2nd matrix

2. Enter matrix as [A]

3. Arrow over to math and choose determinant feature(choice 1)




Determinant of a! 3x3 matrix:
NOTE: its supposed to be a plus sign ! before a 13 it wouldn't let me fix it!


Ex: Find IAI if




Solution:

IAI=0(-1)+2(5)+1(4)=14



Now from this we can tell that determinants are so much fun...but what's even more fun is using them to find a 2X2 inverse!!


Given:
and det A= ad-bc


*switch negate


Guess what? The best part about matrices is... nothing! Haha just kidding! You can use them to solve systems of equations!


Ways:

1) Reduced Row Echelon Form
2) Cramer's Rule
CRAMER'S RULE! because he's just that good!

Let's try an example to show how it works!

2x+3y=17
4x+y= 9


1st: Set up coefficient matrix


x y
IAI= 2-12= -10

2nd: Set up replacement matrices...we know its tedious but keep going, it's worth it in the end!


Replace x column: Replace y column:




3rd- Use Cramer's rule:


Isn't that cool!!!




Applications of Determinants!

You might be thinking where and when will I ever us! e this information in life?
Well, you will be suprised that there are many real life examples that you can amaze your friends with everyday!


1.)Have you ever randomly felt the urge to know the area of an object?

- well now you can easily find this answer on the spot with the matrix knowledge that you have just obtained!


Here's how:

All you need to do is plug the x values of the points into the first column, the y values into the second column and 1 into the third column of the matrix!

2.) You can also use this same method if three points are collinear if the area=0 and the equation of the line passing through a point!


Just look at all of the amazing things you can do with determinants!


PRACTICE PROBLEMS!
Now that you are matrix mast! ers yourselves, you should be anxious to try some problems on ! your own !!!


2.) Let A=Find A^2




3.) Find the inverse of: [4 -1]
[-2 1]





Solve for x and y:






8.) Solve using matrices:

A-4B-3C=-1
2A+3B+7C=5
3A-2B+4C=1

9.) Find the area of the triangle with the vertices (-3,1),! (3,0), and (5,4)


10.) Show, using determinants, that (-2,-5), (1,-2), and (5,2) are collinear.



ANSWERS!

1.)[9 -7]
[-9 4]
2.)[7 -18]
[9 34 ]

3.) [31 12]
[15 11 ]

4.)[2 -4 -6]
[-1 2 3]
[1 -2 -3]
5.) y=2
x=0

6.)x=6 x=-1
7.) 120, 240, and 0 degrees

8.)A=5 B=3 C=-2
9.) 13=area

10.) yes, collinear because area=0

















































determinants of matrices
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