(i) Every entry not on the diagonal of [tex]P^2[/tex] must be 0. For instance, the (1, 2)-th entry in [tex]P^2[/tex] is
[tex]\begin{pmatrix}-\dfrac12&a&a\end{pmatrix}\begin{pmatrix}a\\-\dfrac12\\a\end{pmatrix} = -a+a^2[/tex]
All other non-diagonal entries have the same value. Then
[tex]-a+a^2 = a(a-1) = 0 \implies \boxed{a=0 \text{ or }a=1}[/tex]
Meanwhile, the diagonal entries are
[tex]\begin{pmatrix}-\dfrac12&a&a\end{pmatrix}\begin{pmatrix}-\dfrac12\\a\\a\end{pmatrix} = \dfrac14+2a^2[/tex]
(ii) Since [tex]P^2[/tex] is diagonal, the determinant is simply the product of the entries along the diagonal. If a = 0, these entries are each 1/4, so that
[tex]|P^2| = \left(\dfrac14\right)^3 = \boxed{\dfrac1{64}}[/tex]
If a = 1, they are 9/4, so that
[tex]|P^2| = \left(\dfrac94\right)^3 = \boxed{\dfrac{729}{64}}[/tex]
