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

• Sep 25th 2011, 12:46 AM
Pupil
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.
• Sep 25th 2011, 01:55 AM
FernandoRevilla
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.
• Sep 25th 2011, 06:07 PM
Pupil
Re: How to tell if a matrix is invertible without knowing its determinant?
Quote:

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.
• Sep 25th 2011, 06:18 PM
FernandoRevilla
Re: How to tell if a matrix is invertible without knowing its determinant?
Quote:

Originally Posted by Pupil
Null as in the entire row is a row of zeros, correct?

Right.

Quote:

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

Quote:

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.