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$\begingroup$

An is a (n x n) - matrix defined by:
aij = 1      if       i=j
       -1      if       j = i - 1
        j2     if       i = j + 1
        0      else
Prove by induction that det(An) is n!

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    Have you tried Laplace expansion? – 2017-02-01
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    ^ To that end, have you tried *anything*? Please inform us where, specifically, you get stuck. Also, I hate "Piano Man", Billy. – 2017-02-01
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    There's a small problem: the conditions j = i – 1 and i = j + 1 are the same. So, what's $a_{ij}$? – 2017-02-01
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    should be for j2 j = i + 1 – 2017-02-01

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for n=3 the matrix is $\pmatrix{1 & 2^2 & 0 \\ -1 & 1 & 3^2 \\ 0 & -1 & 1}$

Calculating the det(A3) gives 14 $\neq$ 3!