Learning about matrices.... but WHY!?

Question

I am currently learning about matrices in school, and I've run into a bit of a road block: I know how to perform the operations with matrices and all (addition, multiplication, determinants etc...), but i have no idea about WHY a person would want use them!!

Okay, i do understand when a person would use the addition and subtraction of matrices, and why one would use multiplication of an integer with a matrix, and I also know that they can be used to solve systems of equations, but certain aspects about them really confuse like:

-What is the use of multiplying one matrix to another (i.e. you multiply them, you've got a new matrix, but what do those numbers mean to the user?)?

-What exactly does the determinant represent? (i know how it can be used, but what does the determinant tell you about the matrix?)

Sorry about the vagueness of the questions. Guess I can't exactly word them well, but I basically just want to know exactly how these things function (work-related issue)

Answer 1

this is a great question and it really is something that your teacher should address when you're learning about matrices.

the nicest intuitive way to think of matrices is as a multivariable linear map, for example a 2x2 matrix can map the (x,y)-plane to the (u,v)-plane or something like that. this happens via a special case of matrix multiplication:

[a b][x] = [u]
[c d][y] = [v]

corresponds to the equations u = ax + by, v=cx+dy.

now suppose you have a map from the (x,y)-plane to the (u,v)-plane and another map from the (u,v)-plane to the (s,t)-plane. then you can compose these maps to get a map from the (x,y)-plane to the (s,t)-plane, and you get the matrix for that map by multiplying the two matrices together.

if you think of matrices this way, the determinant has a nice geometric interpretation. look at the unit square in the (x,y)-plane, and then look at its image in the (u,v)-plane (it will always be a parallelogram). you can show that the area of the parallelogram is equal to the determinant of the matrix.

in general, if you take any region in the (x,y)-plane, its image in the (u,v)-plane will be larger (in terms of area) by a factor of the determinant. so the determinant measures how the matrix "stretches" or "shrinks" the plane.

for higher-dimensional matrices (e.g. 3x3), the same idea holds, with volume (or higher-dimensional analogues of volume) instead of area.

Answer 2

A matrix is an equation where the unknown quantities are not written - like a short hand.
Instead of solving
5x+3y-7z=5
2x-9y+3z=2
x+y+z=1
simultaneously, we could write them as a matrix and use matrix algebra to solve, where the first colum is x, the second is y, and the third is z.
[5 3 -7 | 5]
[2 -9 3 | 2]
[1 1 1 | 1]

Answer 3

Matrices can relate two different coordinate systems to each other - i.e. you can view the same vector from a different point of view and the matrix relates the difference between the two points of view. Typically, this would be a 3x3 matrix, while the vector would be represented as a 3x1 matrix. The result would a vector in the new reference frame in a 3x1 format.

This is important in generating computer graphics (as Myra noted) and in orbital mechanics, among other things.

It also becomes important in analyzing the dynamics of a spinning object. Ideally, an object will be spinning about its geometric and prinicpal axis, which makes things like the moment of inertia and angular momentum easy to calculate using algebraic equations. If the spin axis doesn't correspond to the object's principal axis, then the moment of inertia about the spin axis will have to be represented in matrix form - i.e. the object's physical characteristics have to be rotated into a new reference frame using the spin axis.

Answer 4

The study of matrices is quite old. A 3-by-3 magic square appears in Chinese literature dating from as early as 650 BC.[1]

Matrices have a long history of application in solving linear equations. An important Chinese text from between 300 BC and AD 200, The Nine Chapters on the Mathematical Art (Chiu Chang Suan Shu), is the first example of the use of matrix methods to solve simultaneous equations. In the seventh chapter, "Too much and not enough," the concept of a determinant first appears almost 2000 years before its invention by the Japanese mathematician Seki Kowa in 1683 and the German mathematician Gottfried Leibniz in 1693.

Magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs conquered northwestern parts of the Indian subcontinent and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Encyclopedia of the Brethren of Purity (Rasa'il Ihkwan al-Safa); simpler magic squares were known to several earlier Arab mathematicians.

After the development of the theory of determinants by Seki Kowa and Leibniz in the late 17th century, Cramer developed the theory further in the 18th century, presenting Cramer's rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan developed Gauss-Jordan elimination in the 1800s.

The term "matrix" was first coined in 1848 by J. J. Sylvester. Cayley, Hamilton, Grassmann, Frobenius and von Neumann are among the famous mathematicians who have worked on matrix theory.

Olga Taussky-Todd (1906-1995) used matrix theory to investigate an aerodynamic phenomenon called fluttering or aeroelasticity during WWII.

By struggling with the why, I guess you might be searching for it's application. Think about this:

Applications

Encryption
See also: Matrix encryption
Matrices can be used to encrypt numerical data. Encryption is done by multiplying the data matrix with a key matrix. Decryption is done simply by multiplying the encrypted matrix with the inverse of the key.


Computer graphics
See also: Transformation matrix
4×4 transformation matrices are commonly used in computer graphics. The upper left 3×3 portion of a transformation matrix is composed of the new X, Y, and Z axes of the post-transformation coordinate space.


Further reading
A more advanced article on matrices is Matrix theoryat

Answer 5

Your question is excellent!

You can also ask said question in a more unique manner! If plain number application is good enough to arrive at "answers to too many varieties of computing" either by algebra, or geometry, or trigonometry, or co-ordinate geometry, or by differentiation, or by integration...

That plain number application is not dominant today, instead of it "too many variants of basic computing" are taught to users which is technically not needed!

Same problem can be solved in different manners and only the best one among those should be allowed to survive!

Mathematicians have to work hard to grasp and standardise superior manners of computing for different applications!

Technology developers partially does it!


Regards!

Answer 6

if you continued to study about matrices you will then study about sth called adjoint matrix and inverse of the matrix, later you will use the determinent to find the inverse matrix, and the determinent is used also to find the "rank" of the matrix that means the number of non zero rows and columns, that is used to know the number of solutions of a set of equations and so on....
also in higher education you will learn about sth called Eigen values and Eigen vectors they also have to do with the determinent and later you will use it to find the inverse, and "similar" matrices,...
Matrices are also used to find different direction vectors for systems or sth like that.
Matrices are used so much in algebra and analytic math ...
Hope i helped!!!

Answer 7

Describing the quantum mechanics of atomic structure, designing computer game graphics, analyzing relationships, and even plotting complicated dance steps!

Math is important, and as a college student, I am telling you it will not go away. You do use math in everyday situations, especially for balancing budgets, determining GPA, research, determining significance, ect...

Answer 8

Matrices are engineering.

Fluids, Deforming solids, fields of energy, diffusion of matter.

We live in a 3d space with 1d time for a total of 4 dimensions. A one or even 2d phyiscs is simply inadequate.

The general ideas are valid, but need a mathematics capable of describing what is going on in multiple directions simultanously. To do that they use vectors, matrices, and even tensors.

Want to design a great antenna? You need matrices & maxwells equations.
Want to design a climber for the space elevator, or a race car? To predict loading, failure, and performance to the kind of precision to be useful requires matrices.

Want to understand the stock market? You need to be able to track fields of motion in hundreds or thousands of dimensions simultanously.. thats accomplished with matrices.

Want to build powerplants? Fluid dynamics, heat transfer, radiative energy transfer all require matrices.

Want to understand how to compete as a business against businesses that use game theory to support their decision making? You have to be able to use matrices.

Want to design the next computer chip for Intel? The behavior of the gates and materials is described using.. matrices.

Differential Equations, Matrices, and eigens are all very useful. They are the legos with which modern technology builds its skyscrapers.

Determinants are used in eigenvalues and vectors... the fundamentals of principal component analysis. These are the foundation for facial recognition technology (eigenfaces), failure modes of solids, and many other places.

This is like learning how to add subtract multiply and divide in second grade. The teacher didnt say "you need this to make sense out of geometry, trigonometry, algebra, and calculus, and also to get by in your daily life", they just said "trust me, you will need it later".

Answer 9

Dude, just learn it. Not everything in life has a USE! Thats there's such a thing as garbage. Anyways multiplication is used to solve simultanious equations. Just hold on, you'll see there is some use to it.

Answer 10

buddy there are alot of things you learn in school, than after you say why am I learning this? the key is how good you are able to problem solve and follow instructions.