Prove or disprove that if a matrix has the property
$0 \neq |a_{ii}| \leq \sum_{\substack{j=1 \\j \neq i }}|a_{ij}|$
Then Gaussian elimination without pivoting will preserve this property
I have tried to come up with some counterexamples with no real success so I am currently moving forward with the assumption that this is true. I have messed around with trying to show even the first row subtraction will preserve the property, but it got messy and confusing.
I assume that the idea that when calculating the number in the next row (ex $a_{23} =a_{23}-\frac{a_{21}}{a_{11}}a_{13} \implies |\frac{a_{13}}{a_{11}}|\leq 1$ And |$a_{23}| \leq |a_{22}|$) but i have not been able to followthrough