 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/7/b/iii/solution

Use index contraction:

(MN)_{ij} = (δ_{ik} + 2 ϵ_{ik ℓ} a_{ℓ}) a_{k} b_{j} = a_{i} b_{j} + 2 ϵ_{ik ℓ} a_{k} a_{ℓ} b_{j} = N_{ij} .

The last term vanishes because

a_{k} a_{ℓ}

is symmetric in the contracted indices while the Levi-Civita symbol is antisymmetric. Hence

MN = N

 New to topics? Read the docs here!