What is the determinant of the sum of two matrices
$$\det (G + S)$$
where $S$ is all zeros except for a single column of $1$'s?
$$S = \begin{bmatrix} 0 & ... & 0 & 1 & 0 & ... & 0 \\ 0 & ... & 0 & 1 & 0 & ... & 0 \\ \vdots & & \vdots & \vdots & \vdots & & \vdots \\ 0 & ... & 0 & 1 & 0 & ... & 0 \\ \end{bmatrix}$$
I understand this can be solved by breaking up the determinant into columns, but I am unsure of how to do this. Also, $S$ is clearly singular - is there a general rule for the determinant of the sum of a singular and non-singular matrix (i.e. $G$ orthogonal $S$ singular)? Any help greatly appreciated
Im essentially asking what happens to the determinant of a matrix when you add $1$ to each entry in a column. Specifically, I am interested in the case where $G$ is orthogonal with $\det(G) = -1$. I would also be interested in the case where we add $1$ to just a single entry.