Matrices math

Example:
Define a square matrix(just for simplicity) A with a₁ and b₁ in the top row and and a₂ and b₂ in the second row and a in the first columm and bs in the second columm.The determinant of the matrix would be
det(A)=(a₁*b₂-b₁*a₂)
Which you could also write with two vertical lines, like the abs value.

 

One can show that linear equation sytems have solutions for exapmle. It comes n handy all the time. Another example would be that one can write many equation in quantum mechanocs much more elegant that way.

 

There's a theorem that says that the following statements about an arbitrary ##n\times n##-matrix ##A## are equivalent:

(a) ##\det A\neq 0##.
(b) A is invertible
(c) The rows of A are linearly independent elements of ##\mathbb R^n##.
(d) The columns of A are linearly independent elements of ##\mathbb R^n##.

So if you're interested in finding out if any of the statements (b)-(d) is true, you just compute ##\det A##.

The case n=2 is easy to prove. If you define a,b,c,d by ##A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}## you can show that if ##A## has an inverse, it has to be
$$\frac{1}{ad-bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}.$$ So if ##ad-bc## (the determinant of A) is zero, then ##A## isn't invertible.

 

In applications, the determinant can used (in 2-D) to compute the "oriented" area of a parallelogram whose sides are given by 2 row vectors. In higher dimensions, it can be used to compute an "oriented" volume.

Thinking a square matrix M applied as a linear transformation to a vector x, T(x) = Mx, the determinant of M indicates how T expands or contracts volumes (when T is applied to each vector defining a side of a pareallelepiped to produce a transformed parallelepiped).