Proof of an equation about determinant

Question

Please prove an equation about a determinant: (Sorry, I cannot figure any solution)

$$\text{det}(I+ab^T)=1+a^Tb$$

Answer 1

This is simply an application of Sylvester's determinant theorem.

Answer 2

$ I+ab^T $ is the sum between the identity matrix and a matrix with rank $\leq 1$. Since the identity matrix commutes with every matrix, the only eigenvalue of $I$ is $1$ and the only (possibly) non-zero eigenvalue of $ab^T$ is $a^T b$, the eigenvalues of $I+ ab^T$ are $1+ a^T b$ and $1$. The determinant is just the product of the eigenvalues.