Question

Why do we do it by multiplying the Row of first with the Column of second. What’s the practical use of it and who invented it? Logically 4×2 means four times two or two times four. So why can be matrix multiplication just the dot product of corresponding elements?

It’s one of the things that baffles me.

Answer 1

For numbers, 2×4 = 4×2 because they’re commutative. Matrices don’t commute so commutativity of the underlaying numbers really has nothing to do with it.

The idea is that a vector (by which I’ll mean a column vector with the entries written vertically) is an entity in a vector space. This vector space has addition and scalar multiplication defined on it. It also comes with a basis, {e_n}. e_i is just the vector with 1 in the i’th component and 0’s elsewhere. Any vector can be written as a linear combination of the {e_n}. For example, in a 2-dimensional space,

|x_1|       |1|       |0|
|x_2| = x_1 |0| + x_2 |1|

A matrix acts on this vector as a linear transformation and yields a new vector. A linear transformation is just a function, T, with T(x + y) = T(x) + T(y) and c T(x) = T(c x) for any vectors, x and y and any real number c (though we can take it over other fields). So a matrix A acts on a vector x and yields another vector y. A x = y.

|a_11 a_12| |x_1|       |y_2|   |x_1 a_11 + x_2 a_12|
|a_21 a_22| |x_2|   =   |y_1| = |x_1 a_21 + x_2 a_22| 

But we can view the matrix as a set of the vectors made of it’s columns so that’s just the same as

x_11 |a_11| + x_2*|a_12|
     |a_22|       |a_22|

So we’ve re-expressed the definition for the action of a matrix on a vector (m*n matrix times a n*1 matrix) as a linear combination of the columns of the matrix.

This is what allows for us to conflate a matrix with a linear transformation. To express a given linear transformation, T, as a matrix, we just put T(e_i) in the i’th column of the matrix. Call this matrix A_T. Then A_T x = x_1 T(e1) + x_2 T(e2) + … + x_n T(en). But by linearity of T, if x = x_1 e_1 + x_2 e_2 + … + x_n e_n, then T(x) = x_1 T(e_1) + x_2 T(e_2) + … + x_n T(e_n). But this is exactly what we wrote before for A_T. So the law for multiplying a vector by a matrix is required to allow us to represent linear transformations as matrices.

Now let’s consider multiplying general matrices. The idea here is composition of linear functions, that is first do T_1 and then do T_2. That is T_2(T_1(x)) for some vector x. We know from above that we can view these as matrix multiplications. That is
A_T2 (A_T1 x). Let’s look at it in two dimensions because anything else is masochistic and that suffices to get all the ideas across. Let’s relabel the matrices as A_t2 = A and A_T1 = B. Then we have

 A(B x) = |a_11 a_12| (|b_11 b_12| |x_1|)
          |a_21 a_22| (|b_21 b_22| |x_2|)

        = |a_11 a_12| |x_1 b_11 + x_2 b_12| 
          |a_21 a_22| |x_1 b_21 + x_2 b_22|

        = |(x_1 b_11 + x_2 b_12) a_11 + (x_1 b_21 + x_2 b_22) a_12|
          |(x_1 b_11 + x_2 b_12) a_21 + (x_1 b_21 + x_2 b_22) a_22|

        = |x_1 (a_11 b_11 + a_12 b_21) + x_2 (a_11 b_12 + a_12 b_22)|
          |x_1 (a_21 b_11 + a_22 b+21) + x_2 (a_21 b_12 + a_22 b_22)| 

        = |(a_11 b_11 + a_12 b_21) (a_11 b_12 + a_12 b_22)| |x1|
          |(a_21 b_11 + a_22 b+21) (a_21 b_12 + a_22 b_22)| |x2|

Which is just matrix multiplication.

PS. Also probably belongs on Math.SO but I’m not voting to close because I answered. It might be too basic for there as well.