Properties of this rank-$1$ matrix

Question

I've forgotten a lot of basic linear algebra so I was hoping someone might be able to provide a few properties of this matrix.

$$\mathbf b \mathbf a^\top = \begin{pmatrix} a_1b_1 & a_2b_1 & a_3b_1 & \cdots & a_nb_1 \\ a_1b_2 & a_2b_2 & a_3b_2 & \cdots & a_nb_2 \\ a_1b_3 & a_2b_3 & a_3b_3 & \cdots & a_nb_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_1b_n & a_2b_n & a_3b_n & \cdots & a_nb_n \end{pmatrix}$$

I know the determinant is $0$ and the rank is $1$, so the nullity must be $n-1$. Are there any other properties this matrix has?

Thanks in advance.

Answer 1

For example: $det(I + ba^T) = 1 + a^Tb$