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Why can't we multiply corresponding elements like addition is done?
Is there a specific reason why it won't be significant?
By definition, we have to multiply a row by columns.
Why such a definition other than multiplying corresponding elements?
Please ignore my ignorance. I had nowhere to ask. :(
The matrix multiplication is designed in such a way, that one can represent system of linear equations:
$$ \left\{\begin{array}{rcl} a_{11} \cdot x_{1} + a_{12} \cdot x_{2} + \dots + a_{1n} \cdot x_{n} = b_{1} \\ a_{21} \cdot x_{1} + a_{22} \cdot x_{2} + \dots + a_{2n} \cdot x_{n} = b_{2} \\ \dots \\ a_{m1} \cdot x_{1} + a_{m2} \cdot x_{2} + \dots + a_{mn} \cdot x_{n} = b_{m} \end{array}\right. $$
as $A \cdot x = b$, which seems to be quite a natural representation.
You can do such element-wise multiplication of matrices, but it obviously represents a different kind operation. The 'standard' way of multiplying matrices has important applications in linear algebra and, as such, in many areas of science and engineering. The element-wise multiplication has other (typically less used) practical applications.
You can multiply matrices that way, but it turns out to not be as useful in as many cases. Matrix multiplication the way you're taught turns out to be the most useful multiplication-like operation you can do on matrices, so it's the one that just gets the name "product". The other ways you can multiply matrices are not as commonly used, so they get other names, like elementwise (or Hadamard) product, and Kronecker product.
This is a result of what matrices are. There are different interpretations that apply, but here is the one I most often think of:
A matrix is a description of a linear map. It describes this linear map by listing which column vectors the basis vectors are mapped to (the first basis vector is mapped to the first column of the matrix, the second basis vector is mapped to the second column, and so on). And the one multiplication-like operation you can do with linear maps is to compose them, i.e. doing one after the other.
So, if you have two matrices $A, B$, each representing a linear map, then the matrix that represents the composition of the two would deserve the name "the product of $A$ and $B$". And if you work through the geometry and the algebra, the standard matrix product is what falls out.
What do you mean by normal way?
If you mean dot product of Matrics, then yes,
$$ [A_{ij}] \times [B_{ij}] = [A_{ij}\times B_{ij}] $$
But, there are different types of multiplications for matrices, they have some different purposes for each kind of multiplication.
I think other answers have already pointed out the essential.
I would like to give a more philosophical answer to the more general question: how do we give definitions in mathematics?
You can give any definition in mathematics. You just state "I will call Goofy a thing such that A, B and C". There is no problem. Any definition is totally legit.
The problem is: not all definitions are meaningful or useful. A definition is useful if it gives nice and interesting theorems. Indeed in mathematics usually definitions came after theorems were proved. Theorems suggest good definitions and definitions are given according to theorems.
An example to make it clear: how to choose the "correct" definition of prime number from the following:
1) a prime number is a positive integer divisible only by 1 and itself;
2) a prime number is a positive integer with exactly 2 divisors.
The only difference le wheter 1 is prime or not. If you choose 1 to be prime, then the Fundamental Theorem of Arithmetic (the decomposition in prime numbers is unique) fails.
Thus definition 2 is the correct one.
