I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of finding these things but what's the purpose of finding these things? What actually Matrices do which makes it solving equations easier. 

Why determinant is equal to $(ab)-(cd)$ not $(cd)-(ab)$ for the matrix, $\left[\begin{matrix}a&c\\d&b\end{matrix}\right] $ ?

Why inverse of $\rm A$ is equal to $\dfrac{\operatorname{adj}A} {\det A}$ not $\dfrac{\det A} {\operatorname{adj}A}$ ?

I've read many answers like, 
[What is the usefulness of matrices?](https://math.stackexchange.com/questions/160328/what-is-the-usefulness-of-matrices) , they say that matrices do this and that ,but they does not explain how or why ?

Also what is the relation between vectors and matrices?