Eigenvalues of vectors of 2*2 matrix A (2 x 2) matrix $\begin{bmatrix}a & c+id\\c-id & b\end{bmatrix}$ 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 − λI| = det(A − λI) = 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?

The characteristic polynomial is $(\lambda-a)(\lambda-b)-(c^2+d^2)$. Note that the first term already has roots $a,b$, so the discriminant is already positive; $c^2+d^2\ge0$ and increasing $c^2+d^2$ will only increase the discriminant further.
The only possibility for the matrix to have only one eigenvalue is therefore $c=d=0$ and $a=b$, i.e. the matrix is a multiple of the identity matrix.
