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? , 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?
Best Answer
Matrix theory can be viewed as the calculational side of linear algebra. Linear algebra is the theory of vectors, vector spaces, linear transformations between vector spaces, and so on, but if one wants to calculate particular instances, one uses matrix algebra. In part it is a body of notational conventions for how one represents the abstractions described by linear algebra, and in part a collection of recipes for manipulating these notations.
The boundary between MA and LA is not crisp, and a case can be made that there are topics in MA that are not really discussed in LA, so my description above is perhaps simplistic. In particular, the Perron-Frobenius theorem seems inherently bound to matrices and don't seem to have clean abstract linear algebra formulation. Similarly the subject of "total positivity".