why the Identity matrix have 1's at the Main Diagonal Could anyone explain why the 1's in the identity matrix are present in the main diagonal.
$ A=\begin{bmatrix} 4 & 2 \\ 8 & 0 \end{bmatrix} $
$ B=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $
$ C=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $
while I have done the multiplication of A & B and A & C I have got the same result.Is there any other reason for selecting 1's at the main diagonal in I.
 A: The identity matrix is used to express the identity linear operator $id:V\to V$ on some basis $[b_1,\ldots,b_n]$ of the vector space$~V$. In general column $j$ of such a matrix gives the coordinates, in the basis used, of the image under the operator of the basis vector $b_j$. Here the operator maps $b_j$ to itself, so the question is what are the coordinates of $b_j$ in the basis $[b_1,\ldots,b_n]$? Without knowing much about the basis vectors, you can see that the coordinates are $(0,\ldots,0,1,0,\ldots,0)$, where the $1$ is at the position ($j$) of $b_j$ in the list. Now if you put those coordinates in column $j$ of a matrix, the entry $1$ ends up on the main diagonal. Do this for every column, and you get $1$'s on the main diagonal, and $0$'s everywhere else. That's the identity matrix.
Note that this works for any basis, and this is a rather special property of the identity linear operator: you get the same matrix for it, no matter what basis you are using. This is why it is called the identity matrix.
A: It has to play nice with matrix multiplication.

Normally, things are defined like this:
Identity matrix I = 1 on main diagonal
Matrix multiplication L*R takes rows from L and columns from R

This is just an arbitrary convention. There is a way to make every theorem and use of matrices still hold true even with the identity matrix I being defined like C in your example. Call it symmetry world.

In symmetry world, things are defined like this (and math still works!):
Identity matrix I = 1 on off diagonal
Matrix multiplication L*R takes columns from L and rows from R
