Calculate $\lvert A \rvert$ if $a_{ij}=0$ if $i=j$ and $1$ otherwise

Question

Let $n$ be a positive integer and let $A=[a_{ij}] \in M_{n\times n} (R)$ be the matrix defined by

$a_{ij}=0$ if $i=j$
$1$ otherwise

To be honest, I've only calculated determinants of matrices with numbers, nothing like this.

Answer 1

Add all the columns to the first one and then subtract the first row from the others we find

$$\left|\begin{array}\\ n-1&1&\cdots&1\\ 0&-1&\cdots&0\\ \vdots&\cdots&-1&0\\ 0&\cdots&&-1 \end{array}\right|=(-1)^{n-1}(n-1)$$

Answer 2

Another way is to use the fact that the determinant is the product of the eigenvalues. Let $B$ be the matrix with ones everywhere. Then $A=B-I$ it is easy to see that the eigenvalues of $B$ are $n$ with multiplicity $1$ and $0$ with multiplicity $n-1$. Now if $(B-I)v=\lambda v$ then $Bv=(\lambda +1)v$ so the eigenvalues of $B$ are $n-1$ with multiplicity $1$ and $-1$ with multiplicity $n-1$. This gives $(-1)^{n-1}(n-1)$ in agreement with the answer of Sami.