T is linear transformation $$T:\mathbb{R}^3\to \mathbb{R}^{2\times 2}$$
$$T(1,9,9)= \begin{pmatrix} 9 & 0 \\ 12 & 82 \end{pmatrix} $$
$$T(0,1,9)= \begin{pmatrix} 0 & 1 \\ 9 & 81 \end{pmatrix} $$
$$T(0,0,1)= \begin{pmatrix} 9 & 9 \\ 93 & 811 \end{pmatrix} $$
need to find $A\in\text{Im}(T)$ such that $$|A|=-441$$
Soo far I found that: $$T(1,0,0)= \begin{pmatrix} 657 & 639 \\ 6627 & 57745 \end{pmatrix} $$ and $$T(0,1,0)= \begin{pmatrix} -81 & -80\\ -828 & -7218 \end{pmatrix} $$ and we know that : $$T(0,0,1)= \begin{pmatrix} 9 & 9 \\ 93 & 811 \end{pmatrix} $$
You don't need to find $T(1,0,0), T(0,1,0), T(0,0,1)$. Simply solve
$$\left|\alpha T(1,9,9)+\beta T(0,1,9)+\gamma T(0,0,1)\right|=-144$$
for either $\alpha,\beta$ or $\gamma$, and you get all the possible solutions.
If you need just one example of a solution, as opposed to the full space of solutions, it can be simpler still, for example find $\alpha$ such that $$\left|\alpha T(1,9,9)\right|=-144$$