= Solution
A <normal matrix> satisfies $AA^\dagger=A^\dagger A$. Direct multiplication, with all parameters real, gives the <normality criterion for a two-mode shear model>:
$$
AA^\dagger-A^\dagger A=
\begin{pmatrix}a^2-c^2&-ib(a+c)\\ib(a+c)&c^2-a^2\end{pmatrix}.
$$
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 $q=-1$. Hence
$$
\boxed{Re_N=-10\ \text{formally};\qquad\text{no normal case exists for }Re>0.}
$$
The 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.
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