MHT CET · Maths · Matrices
If matrix \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]\) and \(A^{-1}=\alpha I+\beta A\) where \(I\) is a unit matrix of order 2 and \(\alpha, \beta\) are constants, then the value of \(\alpha+\beta+\alpha \beta\) is
- A \(11\)
- B \(-7\)
- C \(7\)
- D \(-11\)
Answer & Solution
Correct Answer
(D) \(-11\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & A^{-1}=\alpha I+\beta A \\ & \Rightarrow\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right]^{-1}=\alpha\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]+\beta\left[\begin{array}{ll}1 & 2 \\ 3 & 5\end{array}\right] \\ & \Rightarrow\left[\begin{array}{cc}-5 & 2 \\ 3 & -1\end{array}\right]=\left[\begin{array}{cc}\alpha+\beta & 2 \beta \\ 3 \beta & \alpha+5 \beta\end{array}\right] \\ & \Rightarrow \alpha=-6, \beta=1\end{aligned}\)
Now \(\alpha+\beta+\alpha \beta=-6+1+(-6) \times 1=-11\)
Now \(\alpha+\beta+\alpha \beta=-6+1+(-6) \times 1=-11\)
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