# Determinant of a matrix

• Jun 2nd 2012, 06:39 AM
crow
Determinant of a matrix
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
• Jun 2nd 2012, 07:37 AM
HallsofIvy
Re: Determinant of a matrix
Quote:

Originally Posted by crow
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

I assume you are referring to a matrix equation of the form Ax= b or a system of equations such that A is the matrix of coefficients (It would have been nice if you had said that!).

If that is correct, yes, the system of equations or matrix equation has a unique solution (and A has an inverse) if and only if det(A) is not 0.

However, it is NOT necessarily true that if the det(A)= 0 there are "infinitely many solutions". If det(A)= 0 then A is not invertible so that the equation Ax= b has either an infinite number of soltutions or no solution, depending upon b. For example, in the one dimensional case, the equation 0x= a has (1) any x as solution if a= 0 or (2) no solution is \$\displaystyle a\ne 0\$.
• Jun 3rd 2012, 02:28 PM
crow
Re: Determinant of a matrix