Question

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

Answer 1

Answer: 2020

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]B=\left[\frac{\alpha }{\beta }\right]$
$AB=B$
$\Rightarrow (A-I)B=O$

$\Rightarrow |A-I|=O$, since $B\ne O$
$\left|\begin{array}{cc}(a-1)& b\\ c& (d-1)\end{array}\right|=0$
$ad-bc=2020$