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

1. 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.

2. 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.

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

Originally Posted by FernandoRevilla
$A\in\mathbb{K}^{n\times n}$ 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.

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

Originally Posted by Pupil
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.