 Solution

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

Every column has

n

entries of magnitude at most one, so

∥ a_{j} ∥^{2} \leq n

. Substituting in part (d) gives

∣ det A ∣ \leq j = 1 \prod n n = n^{n /2} .

The bound comes from the maximum allowed column lengths, together with Hadamard's inequality.

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