1

An integer matrix $M = \begin{bmatrix} -a& b& -c\\ d& e& f\\ g& h& -i \end{bmatrix}$ has determinant $c e g + b f g - c d h + a f h + b d i + a e i$.

Assume $0<a,b,c,d,e,f,g,h<B$ holds.

If you pick such an uniformly random matrix what is the probability that it is singular?

Turbo
  • 6,221
  • @Lovsovs 'integer matrix $M$'. – Turbo Mar 08 '17 at 21:59
  • As B gets to be large, small. – Doug M Mar 08 '17 at 22:03
  • @DougM I think of $B$ as a parameter and so probability has to be in terms of $B$. – Turbo Mar 08 '17 at 22:04
  • $ceg+bfg−cdh+afh+bdi+aei= 0$ with $a\cdots i > 0 \implies cdh=ceg+bfg+afh+bdi+aei$ if we do not choose large enough values of $c,d,h$ relative to the others, it will not be singular.... If were to guess, I would say it is on the order of $\frac {1}{B^3}$ – Doug M Mar 08 '17 at 22:12
  • @DougM compared to general $3\times3$ (which has $\frac1{B^2}$ scaling) we get a better scaling right? – Turbo Mar 08 '17 at 22:19

0 Answers0