cold core (aka cooling flow)

$X$-ray observations suggest that the centres of clusters of galaxies are sometimes colder than the rest of the cluster. What’s the physical reason for these cold cores? Would cold cores show up in the thermal Sunyaev-Zel’dovich effect?

bonus question: Can you show that \(\ln\det(A) = \mathrm{tr}\ln(A)\), or equivalently \(\exp\mathrm{tr}(A) = \mathrm{det}\exp(A)\) for any non-singular square matrix \(A\)?

bonus question 2: What do you think, do you have more or less friends than your friends have on average?