A (2 x 2) matrix $\left[\begin{array}{cc}a& c+id\\ c-id& b\end{array}\right]$ where a, b, c, d are real constants will have two different eigenvalues unless it is a multiple of the identity matrix.

i used this $|A-\lambda I|=det(A-\lambda I)=0$ to prove that it will have 2 eigenvalues. How do i prove that it will only have 1 when it is a multiple of the identity matrix?

i used this $|A-\lambda I|=det(A-\lambda I)=0$ to prove that it will have 2 eigenvalues. How do i prove that it will only have 1 when it is a multiple of the identity matrix?