TITLE: If J is a 101×101 matrix with all entries equal to 1 and let I denote the identity matrix of order 101. Then what is the determinant of J-I?
QUESTION [5 upvotes]: If $J$ is a $101\times 101$ matrix with all entries equal to $1$ and let $I$ denote the identity matrix of order $101$. Then what is the determinant of $J-I$ ?

REPLY [7 votes]: Let $J$ be the $n\times n$ matrix with all entries equal to $1$ and let $I$ be the $n\times n$ identity matrix. 
Furthermore, let $A=-I$ and let $u=v=\left(\begin{array}{c}1\\\vdots\\1\end{array}\right)$. Then $J-I=A+uv^T$, such that by this Lemma:
$$\det(J-I)=\det\left(A+uv^T\right)=\left(1+v^TA^{-1}u\right)\det(A)$$

$A=-I$, such that $\det(A)=(-1)^n$.
$A^{-1}=-I$, such that $A^{-1}u=\left(\begin{array}{c}-1\\\vdots\\-1\end{array}\right)$ and $v^TA^{-1}u=-n$, thus 

$$\det(J-I)=(1-n)(-1)^n$$
and for $n=101$ you get 
$$\det(J-I)=(1-101)(-1)^{101}=100.$$