how is a 'matrix' concept created....in maths? is it just by observation?is maths developed by trail n error?

Answer 1

yes....
a lot of intuition, a good sense of numbers, and yes
trial and error, like everything else in life!!!!!!!!!!!!!!!!!!!!!

Answer 2

The concept of a matrix stems from finding the solution to a system of linear equations. Here's an example:

Suppose I have the following equations:

2x+4y=7
5x+2y=8 solve for x,y

Now I could just use substitution methods to find this or I could write the vector (x,y). If I do, then each equation is a dot product that gives the numbers on the right hand side, in other words:

(2 4) (x,y) =7 for 2x +4y=7
(5 2) (x,y) =8 for 5x +2y=8

Now, if I collect the right hand sides of these equations I can make another vector (7,8). This means the left hand side is some operation M (x,y)= (7,8) where the operator M acts on the vector (x,y) to produce the vector (7,8). This is how a matrix is created, it represents this operation, and we have already written out what M is!

| 2 4 | (x,y) = (7,8) we have found the matrix M
| 5 2 |

Once you have M you can solve for the vector (x,y), I won't go into the method here, but this should be a good example to show how matrices were developed in an easy way. Geometrically speaking the solution to a set of linear equations is the point where all the lines intersect, as our example above shows two lines and the vector (x,y) is the point where they intersect!