 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/p4/d/solution

Use a QR factorization, whose existence is supplied. Since

∣ det Q ∣ = 1

and

R

is triangular,

∣ det A ∣ = ∣ det R ∣ = \prod_{j = 1}^{n} ∣ r_{jj} ∣.

Each diagonal entry is one component of its column, so

∣ r_{jj} ∣ \leq ∥ r_{j} ∥ = ∥ a_{j} ∥

∣ det A ∣ \leq j = 1 \prod n ∥ a_{j} ∥.

The argument includes singular matrices, since a zero diagonal factor makes the determinant zero.

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