I know how to solve a simultaneous equation using matrices. But I don't understand how the answer suddenly come out.
The stuff I don't understand:
1. What is the logic behind matrix multiplication?
I mean, to me, it seems matrix multiplication seems random and arbitrary,
Why not column multiply column, or row multiply row. Is this like a definition of matrix or something?
The idea of multiplication in natural numbers, is number of times you add something to itself. i.e $3\times 3 = 3 + 3 + 3$, or $3$ groups of $3$. What is matrix multiplication??
2.Why can matrices be treated like algebra variables?###
After changing the expression into matrix form, the matrices can be used as if they were variable in algebra. (I think) How does this work?
$$AX = C \implies X = A^(-1) C,$$ and just find $X$. Looks like simple algebra.
3. How does the Gauss Jordan and Cramer method work?
I mean, if I follow the steps, I will somehow get the inverse matrix. And if I check, it seems to be the inverse matrix. How does this work??
I'm sorry if this doesn't make sense, I've tried to ask my friends, and they have no idea what I'm talking about.
You can think of a matrix as a linear function. Given a basis for a vector space (The reals are a trivial example) a matrix can be uniquely defined by a linear function. Matrix multiplication is the way it is, to ensure A.B is the same as making the composite of the two corresponding functions. As for the 2nd part that is only try when A is invertible and it works by multiplying both sides of the equation by the inverse of A. As for the 3rd you can check the proof to see why it works