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    invertible matrix

    I was wondering if its possible to know if a matrix is invertible without row reducing it in attempt to make it I. If the matrix is not nxn, it is never invertible?
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    MHF Contributor Drexel28's Avatar
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    Re: invertible matrix

    Quote Originally Posted by Kuma View Post
    I was wondering if its possible to know if a matrix is invertible without row reducing it in attempt to make it I. If the matrix is not nxn, it is never invertible?
    It doesn't make sense to speak of invertibility for non n\times n matrices. In particular, a matrix A which is n\times m can be thought of a linear transformation A:\mathbb{R}^m\to\mathbb{R}^n and so if A is invertible, then the associated map is an isomorphism, which implies that n=m.
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    Re: invertible matrix

    Quote Originally Posted by Drexel28 View Post
    It doesn't make sense to speak of invertibility for non n\times n matrices. In particular, a matrix A which is n\times m can be thought of a linear transformation A:\mathbb{R}^m\to\mathbb{R}^n and so if A is invertible, then the associated map is an isomorphism, which implies that n=m.
    great explanation, thanks. Now in an nxn case, can you simply tell if a matrix is invertible just by looking at it without trying to use the gauss jordan method to find its inverse?
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    MHF Contributor Drexel28's Avatar
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    Re: invertible matrix

    Quote Originally Posted by Kuma View Post
    great explanation, thanks. Now in an nxn case, can you simply tell if a matrix is invertible just by looking at it without trying to use the gauss jordan method to find its inverse?
    Well, since you liked the last explanation, perhaps one in the same vein. Pretend for a second that a matrix A really is just a linear transformation A:\mathbb{R}^n\to\mathbb{R}^n, then one only has to check any of the following equivalent conditions:


    1) A is injective


    2) A is surjective


    3) A does not have zero as an eigenvalue


    4) \det(A)\ne0


    And there are many more, some of which are very similar to the ones I listed (e.g. 1) is easily seen to be eqiuvalent to the existence of a left inverse).
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    MHF Contributor FernandoRevilla's Avatar
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    Re: invertible matrix

    Quote Originally Posted by Kuma View Post
    If the matrix is not nxn, it is never invertible?
    For A\in\mathbb{K}^{m\times n} we have the concepts of left and right inverse: B\in \mathbb{K}^{n\times m} is a left inverse of A iff BA = I_n and a left-invertible matrix is a matrix with at least one left inverse. Similar considerations for right inverse.
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