The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING n m , consisting of all m -tuples of n n complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING n m will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SING n m is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SING n m . To prove this result we identify precisely the group of symmetries of SING n m . We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case m = 1 , and suggests a general method for determining the symmetries of algebraic varieties.