How do matrices work?

Simply put, a matrix is a table of m columns and n rows in which you place numbers.
The applications are very different, solving lineair systems is a very common one.

 

As I said, one of the most common (and important) uses is that you can use matrixes to solve systems of lineair equations (by, for example, using gaussian elimination or Cramer's rule for square matrices).

To fully understand minors/cofactors, you'll need to know what determinants are, do you?

 

That's an "array" or a "tableau". Any definition of matrices has to include the ability to add and multiply them.

 

Which is why I said "simply put"

 

You can also think of a matrix as an ordered data structure. A matrix often describes a physical state or property of matter.

 

:surprised Have you looked?!

 

Matrices are important because a great many things can be represented as matrices.

The first example people learn is that of a linear transformation, when dealing with vector spaces. A linear transformation T is a function satisfying:

T(αx + βy) = αT(x) + βT(y)

where α and β are scalars. (If the scalars are, for example, real numbers, we say that this is an R-linear transformation)

Linear transformations are important because they respect the indicated algebraic operations.

 

Check the "Linear and Abstract Algebra" forum!