Photo-acoustic Tomography (PAT) is a promising hybrid imaging which simultaneously takes advantage of the rich contrast attributed to optical imaging and the high spatial resolution brought up by ultrasound. We derived the adjoint of a system of coupled equations that describe the propagation of photo-acoustic waves in linear isotropic inhomogeneous and lossy elastic media with the absorption and dispersion following a frequency power law. The adjoint is derived on a continuous domain, and is thus agnostic to the numerical scheme used for solving the forward problem. Using a pseudo-spectral method for solving the elastic forward problem, we will show that the numerical computation of this continuous adjoint matches the algebraic adjoint of the associated forward problem. Finally, we present our numerical results. I invite anyone interested in inverse problems and numerical analysis topics, especially acoustic/elastic problems, to this talk, and your feedback is highly demanded.