Matrices Math: What is a Determinant?

[yuenkf]

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

 

[moriheru]

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.

 

[yuenkf]

why we want to get the determinant?

 

[moriheru]

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.

 

[Fredrik]

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.

 

[Stephen Tashi]

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).