JEE Main 2021 (Online) 17th March Evening ShiftQuestion 1
Let $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $B=\begin{bmatrix}\alpha\\ \beta\end{bmatrix}\neq\begin{bmatrix}0\\0\end{bmatrix}$ such that $AB=B$ and $a+d=2021$, then the value of $ad-bc$ is equal to $\_\_\_\_\_\_$ .
Matrices
Answer$2020$
Solution
$AB=B\Rightarrow(A-I)B=0$ with $B\ne0$, so $\det(A-I)=0$: $(a-1)(d-1)-bc=0$. $ad-bc-(a+d)+1=0\Rightarrow ad-bc=2021-1=2020$.