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Math Help - How to tell if a matrix is invertible without knowing its determinant?

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    How to tell if a matrix is invertible without knowing its determinant?

    We haven't learned what a determinant is in my linear algebra class, but I should know if a matrix is non-singular/invertible in order to infer if it has a unique solution or not for an upcoming midterm. Any methods that do not include finding the matrix's determinant? Thanks in advance.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: How to tell if a matrix is invertible without knowing its determinant?

    A\in\mathbb{K}^{n\times n} is not invertible iff its row echelon form has at least one null row.
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    Re: How to tell if a matrix is invertible without knowing its determinant?

    Quote Originally Posted by FernandoRevilla View Post
    A\in\mathbb{K}^{n\times n} is not invertible iff its row echelon form has at least one null row.
    Null as in the entire row is a row of zeros, correct? That would imply that the homogeneous system has a nontrivial solution, as in infinitely many solutions? Since a system of equations with infinitely many solutions is not invertible.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: How to tell if a matrix is invertible without knowing its determinant?

    Quote Originally Posted by Pupil View Post
    Null as in the entire row is a row of zeros, correct?
    Right.

    That would imply that the homogeneous system has a nontrivial solution,
    Right

    Since a system of equations with infinitely many solutions is not invertible.
    Better: if a system with n equations and n unknowns has infinitely many solutions, the corresponding matrix is not invertible.
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