Matrix and arithmetic sequence

Algebra Level 3

If a 1 , a 2 , a 3 , , a n a_{1} , a_{2} , a_{3} , \cdots , a_{n} is an arithmetic sequence where n n is a perfect square and n > 4 n>4 , then

a 1 a 2 a n a n + 1 a n + 2 a 2 n a n n + 1 a n n + 2 a n = ? \large \begin{vmatrix} a_{1}&a_{2}&\cdots&a_{\sqrt{n}} \\ a_{\sqrt{n}+1}&a_{\sqrt{n}+2}&\cdots&a_{2\sqrt{n}} \\ \vdots&\vdots&&\vdots \\ a_{n-\sqrt{n}+1}&a_{n-\sqrt{n}+2}&\cdots&a_{n} \end{vmatrix} = \ ?


The answer is 0.

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1 solution

Me Myself
Sep 28, 2017

Determinants are invariant to adding a multiple of one row to another.

Notice that any row minus the first row is a multiple of the vector of all ones: (1 1 1 ... 1 1 1). This is true because the row elements are consecutive members of the same arithmatic sequence.

Thus, any three rows are colinear, and so the determinant must be 0.

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