 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/17/a/solution

The displayed matrix is not orthogonal: its first column has squared Euclidean norm three, whereas every column of an orthogonal matrix must have squared length one.

For the general assertion, put

C = A B

A_{ki} A_{kj} = δ_{ij}

and

B_{ki} B_{kj} = δ_{ij}

. The entries of

C

satisfy

C_{ki} C_{kj} = A_{k ℓ} B_{ℓ i} A_{km} B_{mj} = (A_{k ℓ} A_{km}) B_{ℓ i} B_{mj} = δ_{ℓ m} B_{ℓ i} B_{mj} = B_{ℓ i} B_{ℓ j} = δ_{ij} .

This is exactly

C^{T} C = I

, proving that the matrix product

A B

is orthogonal. The Kronecker delta contracts the two middle indices without changing the multiplication order of the matrices.

 New to topics? Read the docs here!