Solution
= Solution
Each <Pauli matrix> has zero diagonal sum and <determinant> $-1$. Direct multiplication gives $\sigma_x^2=\sigma_y^2=\sigma_z^2=I$. Thus all requested quantities, for each $j=x,y,z$, are
$$
\boxed{\det\sigma_j=-1,\qquad \operatorname{tr}\sigma_j=0,\qquad
\sigma_j^2=I,\qquad\sigma_j^{-1}=\sigma_j.}
$$
For example, $\sigma_y^2=\operatorname{diag}((-i)i,i(-i))=I$, checking that its imaginary entries do not introduce a minus sign in the square.