Is it just me, or is the add+sum row by column matrix multiplication system a little contrived? It's obviously useful, but I don't understand why the process of adding and summing is called 'multiplication', and not something else. If I were to intuitively attempt to solve:
|x,2,1||2,1,x|
|3,1,2||3,3,y|
|4,y,z||1,2,3|
If was naive about the correct way to multiply matrices, and knowing that we can multiply by a scalar like so..
|1,0,1| |1x,0x,1x|
x|2,1,2| = |2x,1x,2x|
|4,1,1| |4x,1x,1x|
If I were asked to 'multiply' the two matrices, I would naturally come up with
|x,2,1||2,1,x| |2x,2,1x|
|3,1,2||3,3,y| = |9, 3,2y|
|4,y,z||1,2,3| |4,2y,3z|
As opposed to:
| 2x+7, x+8, , x^2+2y+3 |
| 11, 10, , 3x+(y+6) |
|3y+(z+8), 3y+(2z+4), 4x+(y^2+3z)|
Is there a name for this? Why is the summing of the products of row 1 of A + col 1 of B called 'multiplication' as opposed to the operation of multiplying A(x,x) by B(x,x)?