Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{d}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0\end{array}\right]$ such that
$\mathrm{AB}=\mathrm{B}$ and $\mathrm{a}+\mathrm{d}=2021$, then the value of $\mathrm{ad}-\mathrm{bc}$ is equal to
$A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right], \quad B=\left[\begin{array}{l}\alpha \\ \beta\end{array}\right]$
$\mathrm{AB}=\mathrm{B}$
$\Rightarrow(\mathrm{A}-\mathrm{I}) \mathrm{B}=\mathrm{O}$
$\Rightarrow|\mathrm{A}-\mathrm{I}|=\mathrm{O}$, since $\mathrm{B} \neq \mathrm{O}$
$\left|\begin{array}{cc}(\mathrm{a}-1) & \mathrm{b} \\ \mathrm{c} & (\mathrm{d}-1)\end{array}\right|=0$
$a d-b c=2020$