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JEE Main 2022 · 27 Jul · Shift 1, Mathematics question 4

Let A=(12−2−5)A=\begin{pmatrix}1&2\\-2&-5\end{pmatrix}. Let α,β∈R\alpha,\beta\in\mathbf R be such that αA2+βA=2I\alpha A^2+\beta A=2I. Then α+β\alpha+\beta is equal to

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Answer and worked solution

Official answer: D

Cayley-Hamilton. tr⁡A=−4\operatorname{tr}A=-4, det⁡A=−1\det A=-1, so A2+4A−I=0A^2+4A-I=0, i.e. 2A2+8A=2I2A^2+8A=2I.

α+β=2+8=10\alpha+\beta=2+8=10. (D)

Topics:Matrix algebra
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