Let
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)
and
\(B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \ne \begin{bmatrix} 0 \\ 0 \end{bmatrix}\)
such that \(AB = B\) and \(a + d = 2021\), then the value of \(ad - bc\) is equal to ___________.
\(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\)
and
\(B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \ne \begin{bmatrix} 0 \\ 0 \end{bmatrix}\)
such that \(AB = B\) and \(a + d = 2021\), then the value of \(ad - bc\) is equal to ___________.
Show Hint
Correct Answer: 2020
Solution and Explanation
\(\begin{vmatrix} a-1 & b \\ c & d-1 \end{vmatrix} = (a-1)(d-1) - bc = ad - bc - (a+d) + 1 = 0\).
Hence \(ad - bc = (a + d) - 1 = 2021 - 1 = 2020\).
Answer: 2020
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