I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations easier, but I don't understand why they're the way they are. Why for example are the input data written vertically and the operations horizontally? Wouldn't it be easier if both of them had the same alignment? Did a bunch of mathematicians just decide that this is the way they're gonna be? And also, what's with the determinants? How on earth did we get the formula ad-bc for the 2x2 square matrices? What do they represent, and why is it messy to calculate the determinant for 3x3? And finally, how is it possible to solve system of equations, find volumes, and cross multiply vectors using them? I mean, these things are barely related, yet they can all be solved similarly. Please help me understand all of this, it'd would also be great if someone could provide a material to the history of all of this. Thanks.