Determinant

**kezman** · May 18th 2008, 02:38 PM

A difficult problem for me.
let ( $a_{ij}$ ) = A so that $a_{ij} <= 0$ and $\sum\limits_{j=1}^{n} {a_{ij}}$ $> 0$ . Prove that det A > 0

**KGene** · May 18th 2008, 03:55 PM

Originally Posted by kezman

A difficult problem for me.
let ( $a_{ij}$ ) = A so that $a_{ij} <= 0$ and $\sum\limits_{j=1}^{n} {a_{ij}}$ $> 0$ . Prove that det A > 0

How can these $a_{ij}\leq 0$ and $\sum_{j=1}^na_{ij}>0$ both true simultaneously?

**kezman** · May 18th 2008, 04:08 PM

Sorry I forgot:
the first condition $a_{ij} \leq{0}$ is for $i \neq j$
the second condition is for all $a_{ij}$

**KGene** · May 19th 2008, 12:21 AM

Originally Posted by kezman

Sorry I forgot:
the first condition $a_{ij} \leq{0}$ is for $i \neq j$
the second condition is for all $a_{ij}$

So did you mean $a_{ij}\leq 0$ if $i\neq j$ and $\sum_{1\leq i,j\leq n}a_{ij}>0$ ? But this can't be true, a counterexample,
$\left [\begin{array}{cc} 0&-1\\ -1&3\end{array}\right ]$ where the sum of all entries is 1>0 and $a_{12}=a_{21}=-1\leq 0$ . Please double check again

**Opalg** · May 19th 2008, 02:20 AM

Originally Posted by kezman

A difficult problem for me.
let ( $a_{ij}$ ) = A so that $a_{ij} \leqslant 0$ for $i \neq j$ and $\sum\limits_{j=1}^{n} {a_{ij}}$ $> 0$ for 1≤i≤n. Prove that det A > 0

Such a matrix is called diagonally dominant. The result that you want follows from the Gershgorin circle theorem.

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