 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2013/11/a/solution

- 1

. Direct multiplication gives

σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I

. Thus all requested quantities, for each

j = x, y, z

, are

det σ_{j} = - 1, tr σ_{j} = 0, σ_{j}^{2} = I, σ_{j}^{- 1} = σ_{j} .

For example,

σ_{y}^{2} = diag ((- i) i, i (- i)) = I

, checking that its imaginary entries do not introduce a minus sign in the square.

 New to topics? Read the docs here!