Find determinant of the matrix $P$.

Question

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?

Answer 1

Hint:

Add each row above to the last row, then expand by the last row.

Answer 2

Use row operations to simplify the matrix in an upper triangular form. The form of $P$ is, $$P=\begin{pmatrix} 1 &0&0& ...&1 \\ 0&1&0&...&1\\ \vdots\\ -1&-1&-1&...&1 \end{pmatrix},$$ So carry out the row operation (which doesn't change the determinant) $R_n=R_n+R_k$ for $1 \leq k \leq n-1$. The resulting matrix is of the form $$P'=\begin{pmatrix} 1 &0&0& ...&1 \\ 0&1&0&...&1\\ \vdots\\ 0&0&0&...&1+(n-1) \end{pmatrix},$$ so $\det P=\det P'=1 \cdot 1 \cdot...\cdot1 \cdot n=n$.