Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Question

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} a&b\\c&d\end{bmatrix}$, we get $f(X) =a^2+16c^2+b^2+16d^2$ subject to $g(X)=ad-bc$.

I want to use Lagrange's multiplier method. It says that $Df=\lambda Dg$. I have just one question. Can I consider $f$ to be a function of $a,b,c,d$?? (If yes then I am done) Then only derivative matrix make sense.

If no, then how to proceed??

Thanks for the help!!