Let $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{l}\alpha \\ \beta \end{array}\right]\ne \left[\begin{array}{l}0\\ 0\end{array}\right]$ such that $AB=B$ and $a+d=2021,$ then the value of $ad-bc$ is equal to ______ .

Given $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right], B=\left[\begin{array}{l}\alpha \\ \beta \end{array}\right]$

And, $AB=B$

$\Rightarrow (A-I)B=O$

$\Rightarrow \left|(A-I)B\right|=0$

$\Rightarrow \left|(A-I)\right|\times \left|B\right|=0$

$\Rightarrow |A-I|=0,$ since $B\ne 0$

$\Rightarrow \left|\begin{array}{cc}(a-1)& b\\ c& (d-1)\end{array}\right|=0$

$\Rightarrow \left(a-1\right)\left(d-1\right)-bc=0$

$\Rightarrow ad-a-d+1-bc=0$

$\Rightarrow ad-bc=a+d-1$

$\Rightarrow ad-bc=2021-1=2020.$