What are the implications of calculating the determinant of a matrix?

[knightmetal]

Hello,

I'd like to know if the following two paragraphs regarding the determinant of a matrix are correct and also, am I missing any other important implications by calculating the determinant? any other important things I can find from with that value? thanks.

1. If det A=0 <=> Linear Dependence <=> Infinitely many solutions (hence non trivial solution) <=> non invertible (or singular) matrix <=> vectors are parallel.

2. If det A != 0 <=> L.I <=> unique solution <=> invertible (also nonsingular or regular) matrix

 

[DonAntonio]

Hello,

I'd like to know if the following two paragraphs regarding the determinant of a matrix are correct and also, am I missing any other important implications by calculating the determinant? any other important things I can find from with that value? thanks.

1. If det A=0 <=> Linear Dependence <=> Infinitely many solutions (hence non trivial solution) <=> non invertible (or singular) matrix <=> vectors are parallel.

2. If det A != 0 <=> L.I <=> unique solution <=> invertible (also nonsingular or regular) matrix

It's hard to be certain about what you mean: what "vectors" are parallel? Do you mean the matrix's rows (columns)? Then this is false

if the matrix is n x n , n > 2, as what is actually true is that at least one row is in the span of the other ones.

This determinant-not-zero thing is one of the basic mathematics ideas with more equivalent formulations: you could also say that det A = 0

iff ker A = 0 iff A is onto (when seen as linear operator) iff zero is NOT one of its eigenvalues iff for any non-zero b, the system Ax = b has one single solutions...

DonAntonio

 

[Vargo]

Aside from what DonAntonio mentioned, one important fact you missed is that the determinant is equal to the volume of the parallelepiped spanned by the columns (or the rows). See http://en.wikipedia.org/wiki/Determinant#Volume_and_Jacobian_determinant and scroll down to the colorful pictures.

This explains (geometrically) why a determinant of zero corresponds to a singular matrix. It also goes a long way toward explaining the rule for changing variables in multiple integrals.

 

[algebrat]

the opposite of one solution is zero or at least two.

if det A=0, it is possible to have no solution. for instance 0x=0 has many solutions, while 0x=1 has none.

In other words, if det A =0, then A is not injective nor surjectinve.

 

[knightmetal]

Hey guys thanks a lot for your replies, it helped :)

 

[NegativeDept]

am I missing any other important implications by calculating the determinant? any other important things I can find from with that value?

Another useful thing to know: many numerical algorithms run into horrible problems whenever $\textrm{Det}[\hat{M}] \approx 0$. (E.g. matrix inversion is unstable.)

There's also a fairly comprehensive list of singular-matrix implications on Wikipedia:
http://en.wikipedia.org/wiki/Singular_matrix#Properties

 

[mathwonk]

heres all there is to know about determinants.