Q: The determinant of a matrix $A \in \mathbb{R}^{n \times n}$?

I really struggle with this problem, how do you calculate the determinant of matrix $A \in \mathbb{R}^{n \times n}$, whose expression is $$\begin{pmatrix} 2 & 1& ...& 1\\ 1& 2& ...& 1\\ 1& 1& 2& ...\\ 1& 1& ...& 2 \end{pmatrix} ?$$

-
Did you try some induction? –  DonAntonio Feb 2 '14 at 13:35
A more general problem which can be solved by very similar methods: math.stackexchange.com/questions/86644/… and math.stackexchange.com/questions/382799/… –  Martin Sleziak Dec 16 '14 at 15:46

Hint: Find all the nonzero eigenvalues of $A - I$ with their multiplicities and recall that commuting matrices are simultaneously triangularizable.
I don't get the simultaneously triangularizable part. Certainly one does not need that rather deep fact to see that the eigenvalues (with algebraic multiplicities) of $A-I$ are obtained from those of $A$ by subtracting $1$ from each eigenvalue? –  Marc van Leeuwen Feb 2 '14 at 13:58