Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 &\ldots & 2 \end{pmatrix} $$
For $n=2$, I have$$ \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} $$
Then $det = 3$.
For $n=3$, we have $$ \begin{pmatrix} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \\ \end{pmatrix} $$
Then $det = 4$.
For $n=4$ again we have
$$ \begin{pmatrix} 2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1\\ 1 & 1 & 2 & 1\\ 1 & 1 & 1 & 2 \end{pmatrix} $$ Then $det = 5$
How can I prove that the determinant of nun matrix is $n+1$.