This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few):
- Eigenvalues of a matrix of $1$'s
- Eigenvalues for the rank one matrix $uv^T$
- Calculating $\det(A+I)$ for matrix $A$ defined by products
- How to calculate the following determinants (all ones, minus $I$)
- Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$)
- Determinant of a special $n\times n$ matrix
- Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal
- Determinant of a matrix with $t$ in all off-diagonal entries.
- Characteristic polynomial - using rank?
- Determinant of rank-one perturbations of (invertible) matrices
The general question of this type is
Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?
A clearly very closely related question is
What is the characteristic polynomial of a matrix $A$ of rank$~1$?