Some quick googling finds the Julia benevolent overlords had already gotten you covered almost 10 years ago
https://github.com/JuliaLang/julia/issues/659
using BenchmarkTools
a = rand(100,100); b = rand(100,100);
@benchmark $a' * $a
# median time: 110.190 μs (0.00% GC)
@benchmark $b' * $a
# median time: 836.717 μs (0.00% GC)