Consider $J$ to be an $n\times n$ matrix whose entries are all $1s$ .
If $P$ is an $n\times n$ matrix such that
$P=$ \begin{bmatrix} v_1| v_2|,\ldots |v_{n-1} |v_n\end{bmatrix}
where the columns $v_i=e_i-e_n$ for $1\le i\le n-1$ and $v_n=\sum e_i$ where $e_i$ is the ith column of the Identity Matrix
Note that $v_i,1\le i\le i-1$ are the eigen vectors corresponding to $0$ of $J$ and $v_n$ is an eigen vector corresponding to $n$ of $J$
Find $\det P$.
It is very difficult to expand by Laplace Expansion
Is there any efficient way to to find the determinant?
Best Answer
Hint:
Add each row above to the last row, then expand by the last row.