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Lots of nonsingular matrices

Source: Brazilian Math Olympiad 2005 Undergrad problem 5

October 24, 2005
linear algebramatrixcalculusintegrationMITcollege

Problem Statement

Prove that for any natural numbers 0≤i1<i2<⋯<ik0 \leq i_1 < i_2 < \cdots < i_k and 0≤j1<j2<⋯<jk0 \leq j_1 < j_2 < \cdots < j_k, the matrix A=(ars)1≤r,s≤kA = (a_{rs})_{1\leq r,s\leq k}, ars=(ir+jsir)=(ir+js)!ir! js!a_{rs} = {i_r + j_s\choose i_r} = {(i_r + j_s)!\over i_r!\, j_s!} (1≤r,s≤k1\leq r,s\leq k) is nonsingular.
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