Determinant of identity matrix minus exponential matrix $\det(I-k \exp (M)),$

Question

I am currently struggling with the following determinant $$\det(I-k \exp (M)),$$ where $I$ is the $2\times2$ identity matrix, $M$ is a $2\times2$ matrix and $k$ is an arbitrary constant. Is there a general way of handling such objects?

Answer 1

I used Wolfram Mathematica code

Det@MatrixFunction[1-k*Exp[#]&,{{a,b},{c,d}}]//FullSimplify//TeXForm

which returns the result

$$-2 k e^{\frac{a+d}{2}} \cosh \left(\frac{1}{2} \sqrt{(a-d)^2+4 b c}\right)+k^2 e^{a+d}+1$$

and I have no reason to doubt its validity. As you can see, the Mathematica function MatrixFunction does the hardest part of the job.

You can check the validity for diagonal matrices with

Det@MatrixFunction[1-k*Exp[#]&,{{a,0},{0,d}}]//Factor//TeXForm

which returns $$\left(e^a k-1\right) \left(e^d k-1\right)$$ as it should.