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.

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

$\displaystyle A\in\mathbb{K}^{n\times n}$ is not invertible iff its row echelon form has at least one null row.

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

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Originally Posted by

**FernandoRevilla** $\displaystyle A\in\mathbb{K}^{n\times n}$ 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.

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

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Originally Posted by

**Pupil** Null as in the entire row is a row of zeros, correct?

Right.

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

Right

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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.