Solution
= Solution
Taking <determinants> in $MN=N$ gives $(\det M-1)\det N=0$. Here $\det M-1=4|\mathbf a|^2>0$ because $\mathbf a\ne0$, so $\boxed{\det N=0}$. This also agrees with $N=\mathbf a\mathbf b^T$ being an <outer product> of rank one.
= Solution
Taking <determinants> in $MN=N$ gives $(\det M-1)\det N=0$. Here $\det M-1=4|\mathbf a|^2>0$ because $\mathbf a\ne0$, so $\boxed{\det N=0}$. This also agrees with $N=\mathbf a\mathbf b^T$ being an <outer product> of rank one.