Past exam of the mathematical methods I course of the University of Oxford / 2013 / 11 / a / 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 $j=x,y,z$, areFor example, ${\sigma }_y^2=\mathrm{diag}((-i)i,i(-i))=I$, checking that its imaginary entries do not introduce a minus sign in the square.