What is the determinant of the matrix

Question

If M be a $2\times2$ matrix such that $$M=\begin{bmatrix}1&\frac{1-i}{\sqrt{2}}\\ \frac{1+i}{\sqrt{2}}&0 \end{bmatrix}$$ then what is the determinant of $e^M$. I think first to calculate $e^M$ then find out the determinant. Am I correct? Pls answer me in detail.

Answer 1

Upon diagonalization of $$M=\begin{bmatrix}1&\frac{1-i}{\sqrt{2}}\\ \frac{1+i}{\sqrt{2}}&0 \end{bmatrix}$$

We get $M=PDP^{-1}$ where $D$ is a diagonal matrix whose diagonal elements are eigenvalues of $M.$

$$ \det (exp (M)) = \det ( exp (D)) =e^{\lambda _1 + \lambda_2}= e^{tr D}=e^{tr M} =e$$

Answer 2

No need:$$\det\bigl(\exp(M)\bigr)=\exp(\operatorname{tr}M)=e^1=e.$$