Question

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

Answer 1

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

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