# Matrices math

• yuenkf

#### yuenkf

1. sorry. erm.. what does the determinant means or functions in matrices , math? thanks..

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.

why we want to get the determinant?

One can show that linear equation systems 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.

why we want to get the determinant?
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.

why we want to get the determinant?

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).

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