Chapter 11 Matrices and Trigonometry
Joel
Discrete Math
11.1
A point matrix is a 2 x 1 matrix
Given that, a n-gon matrix is a 2 x n matrix.
Ex.1 If triangle ACD had the coordinates A(2,1), C(4,4), and D(5,2) it would look like this on a graph.
A C D
[ 2 4 5]
[ 1 4 2]
rx = [ 1 0] ry = [ -1 0] ry=x = [ 0 1]
[ 0 -1] ; [ 0 1] [1 0]
Ex.2 if multiplied by triangle ACD the results would be:
[ 2 4 5] [ -2 -4 -5] ; [1 4 2]
[ -1 -4 -2] [ 1 4 2] [ 2 4 5]
SIZE CHANGES
if multiplied by [ x 0] gives an multplication of x
[ 0 x ]
[ 5 0] multiplied by triangle ACD is [ 10 20 25]
[ 0 5 ] [ 5 20 10]
11.3
A composite is the multiplication of reflections or rotations to make a new image. It also shows how it is multiplied.
basic rotations:
R90 = [ 0 -1] R180 = [ -1 0] R270 = [ 0 1]
[ 1 0] [ 0 -1] ; [-1 0]
If the rotations are forgotten the matrix basis theorem comes in to play:
the 1st colunm is (1, 0) and the 2nd column is (0,1) now think of a rotation or look at the rotation given. Ex- 180 degrees, think about what 180 degrees does. (It turns things the opposite way. Think about what it does to both the x and the y. So this is what it would look like: (1, 0) changes to (-1,0) because of the x axis, and (0,1) becomes (0,-1) because of the y axis.
11.7
[ 1 0] = identity matrix
[0 1]
Definition of inverse matrix: If there are two matrices that map out the same points then they are both inverses, but it has to be commutative. If the matrices product is the identity matrix then the matrices are inverses.
Only square matrices can be inverses
Ex.1
Using a graphing calculator, look and find a function x-1( the -1 will be like an exponent). press that after the matrix and their will be an inverse.
Ex. 2
Switch and flip - Ms. Blackwell's invention
[ 1817 2055] = [ 5023 -2055]
[ -6112 5023] [ 6112 1817]
Switch the top left with the bottom right and change the signs of the top right and bottom left.
Ex.3
What is the inverse matrices? BY HAND
[ 2 9] [ a b]
[4 3] [ c d]
2a + 9c = 1 2b + 9d = 0
4a + 3c = 0 4b + 3d = 1
multiply 2a + 9c = 1 by -2 and add it to the other equation. Cancel out a, and solve for c. C should equal 2/15. Take 4b + 3d = 1 and multiply it by -3 then add it to the other equation. Cancel out the d and solve for b. B should equal 3/10. Plug c back into the equation and a should come out as -0.1. Plug b back into the equation and d should come out as -0.066. Check by multiply the new matrix by the original, it should come out to the identity matrix.
Kindness is wisdom.
Phillip J. Bailey merlystreeponline.net/qp1.htm
Kindness is the character trait displayed saying that to have wisdom one has to be kind also0