<s>[INST] Is the determinant of this matrix positive or negative? $\left( \begin{array}{ccc}
1 & 1000 & 2 & 3 &4\\
5 & 6 &7&1000 &8\\
1000&9&8&7&6\\
5 & 4&3&2&1000\\
1&2&1000&3&4\\ \end{array} \right)$
When I compute the determinant online, I find that it is positive, but I'm supposed to "see" something about the matrix that allows me to know the determinant is positive. What properties does this specific matrix have that allow you to deduce the determinant will be positive?
 [/INST] By definition, the determinant is the sum of $60$ positive and $60$ negative terms, one of which is $\mbox{sgn}{\left(
\begin{array}{ccccc}
 1 & 2 & 3 & 4 & 5 \\
 3 & 1 & 5 & 2 & 4 \\
\end{array}
\right)}(1000)^5=\mbox{sgn}\left((13542)\right)(1000)^5=+(1000)^5$.
Each of the other terms has absolute value at most $9\cdot1000^4$, so the sum of the 60 negative terms is greater than or equal to $-60\cdot9\cdot1000^4$, and the determinant is greater than or equal to $(1000)^5-60\cdot9\cdot1000^4$, which is positive.
</s>