JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
Let \(\alpha\) and \(\beta\) be real numbers. Consider a \(3 \times 3\) matrix \(A\) such that \(A ^2=3 A +\alpha I\). If \(A ^4=21 A +\beta I\), then
- A \(\alpha=1\)
- B \(\alpha=4\)
- C \(\beta=8\)
- D \(\beta=-8\)
Answer & Solution
Correct Answer
(D) \(\beta=-8\)
Step-by-step Solution
Detailed explanation
\(A ^2=3 A +\alpha I\) \(A ^3=3 A ^2+\alpha A\) \(A ^3=3(3 A +\alpha I )+\alpha A\) \(A ^3=9 A +\alpha A +3 \alpha I\) \(A ^4=(9+\alpha) A ^2+3 \alpha A\) \(=(9+\alpha)(3 A +\alpha I )+3 \alpha A\) \(= A (27+6 \alpha)+\alpha(9+\alpha)\)…
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