Solve the Following Questions
Question:

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

Solution:

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

$\mathrm{AB}=\mathrm{B}$

$\Rightarrow(\mathrm{A}-\mathrm{I}) \mathrm{B}=\mathrm{O}$

$\Rightarrow|\mathrm{A}-\mathrm{I}|=\mathrm{O}$, since $\mathrm{B} \neq \mathrm{O}$

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

$a d-b c=2020$

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