# Help with a solution

by Perrry
Tags: solution
 P: n/a Let $$\begin{gather*}A_n\end{gather*}$$ be an nxn matrix with the matrixelement $$\begin{gather*}a_ik\end{gather*}$$=i+k, i, k = 1, ... ,n. Decide for every value the n-determinant $$\begin{gather*}D_n\end{gather*}$$ = det($$\begin{gather*}A_n\end{gather*}$$). Donīt forget the value of n=1. We are two guys here at home that donīt get it right. What shall we start with? We are both newbies on this! Thanks in advance Perrry
 P: 16 Consider a matrix $$A_n$$ such as the one you described with $$n \geq 3$$. Look at the columns of $$A_n$$: the first column, $$c_1$$ looks like: $$c_1=(2,...,n+1)^T$$ the next column looks like $$c_2=(3,...,n+2)^T$$ and the third column looks like $$c_3=(4,...,n+3)^T$$. Since we took $$n \geq 3$$ we know that we can always get $$c_1,c_2,c_3$$. Observe that: $$c_3-c_2=c_2-c_1=(1,...,1)^T$$ Therefore, since $$c_3+c_1=2c_2$$ we have that the columns are linearly dependent. Does this say anything about the determinant?

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