It is known that the mortality problem for 3×3 matrices with integer entries is undecidable. Is the mortality problem for 3×3 orthogonal matrices still undecidable ?
The mortality problem: determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix.
Orthogonal matrices are invertible, therefore it's a fairly easy property to check: it never happens.