 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2008/a/2/a/solution

An invertible matrix is exactly an ordered basis of its column space. The first column has

q^{n} - 1

choices. After

j

independent columns have been chosen, their span has

q^{j}

elements, so the next column has

q^{n} - q^{j}

choices. Therefore

∣ GL (n, q) ∣ = j = 0 \prod n - 1 (q^{n} - q^{j}) .

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