How do I solve this? Find determinant of a matrix

Question

I have a matrix with n rows and n columns: $A=(a_{j,k})$ is defined: $$(a_{j,k})=\begin{cases} \begin{split} 2j (j\neq k) \\ 0(j=k) \end{split} \end{cases}$$
And I need to find $det(A)$. I think that I need to use gauss elimination but I have no idea how to do this here. can anyone help please? thanks.

Answer 1

Taking out the $2j$ factors as suggested leaves $E-I$, with $I$ the identity matrix and $E$ all $1s$. If $a$ has shape $n\times n$ the eigenvalues of $E$ are $0,\,n$, of respective multiplicity $n-1,\,1$. Then $$\det (E-I)=(-1)^{n-1}(n-1),\,\det a=-n!(n-1)(-2)^n.$$

Answer 2

After factoring out $2j$ you will get the matrix $$ \begin{bmatrix}1\\1\\\vdots\\1\end{bmatrix}\begin{bmatrix}1 &1&\ldots&1\end{bmatrix}-I=ee^T-I. $$ Now use the Sylvester determinant formula $$ \det(ee^T-I)=(-1)^n\det(I-ee^T)=(-1)^n(1-e^Te)=(-1)^n(1-n). $$