Find $det(xy^T)$ where $x$, $y$ are vectors from $R^n$, $n$>1

Question

I represented $x$ as $[x_1\ x_2\ ... x_n]^T$, and $y^T$ as $[y_1\ y_2\ ... y_n]$.

Multiplying them produces a matrix $n$x$n$: $$ \begin{pmatrix}x_1y_1&x_1y_2&\dots& x_1y_n\\ x_2y_1&x_2y_2&\dots& x_2y_n\\ \\ \\ \\ \\ x_ny_1& x_ny_2&\dots& x_ny_n \end{pmatrix}$$

(I apologize, I don't know how to format a matrix using LaTeX). :)

Obviously, for $n=2$, $n=3$ the determinant is $0$. I just don't know how to prove it for $n$.

Answer 1

Hint. when both $x,y$ are nonzero, this matrix has rank 1, so cannot be invertible when $n>1$.