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Math Help - Determinant of a matrix

  1. #1
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    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
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  2. #2
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    Re: Determinant of a matrix

    Quote Originally Posted by crow View Post
    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 a\ne 0.
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  3. #3
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    Re: Determinant of a matrix

    Hello HallsofIvy, thanks a lot for your reply, it helped
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