martes, 10 de marzo de 2009

Inverse Matrix

What is it used for?

One of the applications of inverse matrix is in solving simultaneous equations.

If you are good with algebra, you will discover that this inverse matrix way of handling the solution of simultaneous equations is similar, except that they are done as a group, collectively.
The answers to the unknown variables are obtained at one go with this Inverse matrix method.

However, what do you need to know in order to use this Inverse matrix solving?

You have to understand:
1) Convertion of simultaneous equations into a set of matrices
2) Determinant and technique to get its numerical value
3) Minor of the individual elements within the matrix
4) Co-factor of this determinant formed with this set of matrices
5) Transpose of matrix
6) Adjoint matrix obtained with the co-factors and transposed matrix
7) Formula to relate determinant with the adjoint matrix ==> Inverse matrix
8) Matrices multiplication

The list looks amazingly long for matrix novice, but, DO NOT FEAR!

Matrices consist of numbers only, and simple mathematical operations, nothing abstract.
(The details are not presented here for fear that you will leave this site.)

Slowly research into the above terms and see for yourself that they are "friends" and not "foes".

Happy start to matrices and its application.

Basic Trigonometric

Trigonometric Identities - Basic Identities

Trigonometric identities are specific equalities that express one trig function in terms of other trig functions. They are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.

The basic identities are traditionally visualized with a triangle formed by a radius r, length x, and height y:
The basic trig definitions can easily be seen:
Sin(theta) = y/r..... opposite/hypotenuse
Cos(theta) = x/r..... adjacent/hypotenuse
Tan(theta) = y/x..... opposite/adjacent

If we now apply the Theorem of Pythagoras, we can see:
r^2 = x^2 + y^2
Dividing everything by r^2 gives:
1 = (x^2)/(r^2) + (y^2)/(r^2)
1 = (x/r)^2 + (y/r)^2
And then, subbing in the basic definitions, we get:
1 = [Cos(theta)]^2 + [Sin(theta)]^2

And that is the first basic identity. Nothing to it. It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it in proper notation to clean it up a bit... (Blogger is a pain with superscripts and fonts)

Another basic relationship starts with:
Tan(theta) = y/x
But, then sub in the Sine and Cosine definitions (isolated for x and y, respectively) to give
Tan(theta) = (r*Sin(theta)) / (r*Cos(theta))
Tan(theta) = Sin(theta) / Cos(theta)

And that's it again. This is called the Ratio Identity:

Those are now two of the simplest trig identities from which most of the others can be derived.