Determining the n-Determinant for an nxn Matrix with a Specific Matrix Element

[Perrry]

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($$\begin{gather*}A_n\end{gather*}$$). Don´t forget the value of n=1.

We are two guys here at home that don´t get it right. What shall we start with? We are both newbies on this!

Thanks in advance

Perrry

 

[marcmtlca]

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?