Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 4 / b / iv / Solution

A normal matrix satisfies $A{A}^{†}=A^{†}A$. Direct multiplication, with all parameters real, gives the normality criterion for a two-mode shear model:For $a=1$, $b=c=q=10/Re$, both $1-q^2=0$ and $q(1+q)=0$ are required. Their only common solution is $q=-1$. HenceThe original PDF really has these signs and $b=c=10/Re$. Thus parts (v) and (vi) have no physical positive-Reynolds-number case as printed. We give their algebraic continuation at $Re=-10$ below, without silently changing the matrix. The tempting value $Re=10$ makes the off-diagonal magnitudes equal but fails the off-diagonal commutator condition.