This question aims to create an “abstract duplicate” of numerous questions that ask about determinants of specific matrices (I may have missed a few):

- Characteristic polynomial of a matrix of 1‘s
- Eigenvalues of the rank one matrix uvT
- Calculating det(A+I) for matrix A defined by products
- How to calculate the determinant of all-ones matrix minus the identity?
- Determinant of a specially structured matrix (a‘s on the diagonal, all other entries equal to b)
- Determinant of a special n×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?
- Caclulate XA(x) and mA(x) of a matrix A∈Cn×n:aij=i⋅j
- 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 λ a scalar. What is the determinant of A+λI?

A clearly very closely related question is

What is the characteristic polynomial of a matrix A of rank 1?

**Answer**

The formulation in terms of the characteristic polynomial leads immediately to an easy answer. For once one uses knowledge about the eigenvalues to find the characteristic polynomial instead of the other way around. Since A has rank 1, the kernel of the associated linear operator has dimension n−1 (where n is the size of the matrix), so there is (unless n=1) an eigenvalue 0 with geometric multiplicity n−1. The algebraic multiplicity of 0 as eigenvalue is then at least n−1, so Xn−1 divides the characteristic polynomial χA, and χA=Xn−cXn−1 for some constant c. In fact c is the trace \def\tr{\operatorname{tr}}\tr(A) of~A, since this holds for the coefficient of X^{n-1} of *any* square matrix of size~n. So the answer to the second question is

The characteristic polynomial of an n\times n matrix A of rank~1 is X^n-cX^{n-1}=X^{n-1}(X-c), where c=\tr(A).

The nonzero vectors in the 1-dimensional image of~A are eigenvectors for the eigenvalue~c, in other words A-cI is zero on the image of~A, which implies that X(X-c) is an annihilating polynomial for~A. Therefore

The minimal polynomial of an n\times n matrix A of rank~1 with n>1 is X(X-c), where c=\tr(A). In particular a rank~1 square matrix A of size n>1 is diagonalisable if and only if \tr(A)\neq0.

See also this question.

For the first question we get from this (replacing A by -A, which is also of rank~1)

For a matrix A of rank~1 one has \det(A+\lambda I)=\lambda^{n-1}(\lambda+c), where c=\tr(A).

In particular, for an n\times n matrix with diagonal entries all equal to~a and off-diagonal entries all equal to~b (which is the most popular special case of a linear combination of a scalar and a rank-one matrix) one finds (using for A the all-b matrix, and \lambda=a-b) as determinant (a-b)^{n-1}(a+(n-1)b).

**Attribution***Source : Link , Question Author : Marc van Leeuwen , Answer Author : Community*