How can I calculate this determinant? Please can you give me some hints to deal with this :
$\displaystyle \text{Let } a_1, a_2, ..., a_n \in \mathbb{R}$
$\displaystyle \text{ Calculate } \det A \text{ where }$ $\displaystyle A=(a_{ij})_{1\leqslant i,j\leqslant n} \text{ and }$ $\displaystyle \lbrace_{\alpha_{ij}=0,\text{ otherwise}}^{\alpha_{ij}=a_i,\text{ for } i+j=n+1}$

As @MarianoSuárez-Alvarez said, write down the matrix for small $n$ and see how it looks like.
Your matrix is just an anti-diagonal matrix, whose determinant is given by $(-1)^{\frac{n(n-1)}{2}}\prod_{i=1}^n a_i$. 
The sign part $(-1)^{\frac{n(n-1)}{2}}$ comes from arranging the rows to achieve a diagonal matrix whose determinant is just the product of all elements on the diagonal. Swapping two adjacent lines will change the sign of the determinant.
