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**Matrices, Transposes, and Inverses. --- Help?**

Say, you have Matrix A, Y, and a matrix of coefficients C not neccissarily square, and A * C = Y. To solve this, i would think to multiply both sides by the inverse of A, but since it's not necessarily square that won't work. I've seen a solution that im having some questions about. It multiplies both sides by ATranspose, then it multiplies both sides by (A * ATranspose) ^-1. I think i understand that multiplication by ATranspose, so as to get it into a square matrix. Then, i guess i understand the multiplication by the inverse of it, but i think i read somewhere that it is particularly easier to take the inverse of A * Atranspose. I need the simplest method possible, because i have to code a program to solve this equation for me, in C++. I can figure out the code, if i can find a step by step process to finding the inverse, but im really confused regarding taking inverses in general, and if it's easier if you multiply it by the transpose first.

I'd appreciate any help, and, if possible, i need this kinda soon, but even if the deadline passes, i still want to understand, for myself how it's done.