# 2012-6 Matrix modulo p

Let p be a prime number and let n be a positive integer. Let $$A=\left( \binom{i+j-2}{i-1}\right)_{1\le i\le p^n, 1\le j\le p^n}$$ be a $$p^n \times p^n$$ matrix. Prove that $$A^3 \equiv I \pmod p$$, where I is the $$p^n \times p^n$$ identity matrix.

GD Star Rating