How to find the dominant eigenvalue of a Next generation matrix

I have been working with a compartmental model and I am now trying to compute the basic reproduction number. To do this I must find the spectral radius (dominant eigenvalue) of the following matrix.

$$K=begin{bmatrix}frac{beta_{HH}( pi_H/ mu_H)}{(alpha + gamma + mu_H)(mu_H)}&frac{beta_{TH}(pi_H/mu_H)}{mu_T(mu_T * IT)}&frac{beta_{CH}(pi_H/mu_H)}{mu_C(mu_C * IC)}\0&frac{(beta_{TTV} + beta_{TTH})(pi_T/mu_T)}{mu_T(mu_T * IT)}&frac{beta_{CT}(pi_T/mu_T)}{mu_C(mu_C*IC)}\0&frac{beta_{TC}(pi_C/mu_C)}{mu_T(mu_T * IT)}&0end{bmatrix}$$

Next Generation Matrix K

I understand this is done by taking the determinant of $K – lambda I$. I have found the determinant but cannot correctly compute the eigenvalues. I know when using a $2 times 2$ Next Generation Matrix there is an easy formula to find the basic reproduction number but I am unsure for a $3 times 3$. If anyone can give their insight that would be great. Thank you.