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JEE Main 201815 AprilEvening ShiftOnline
Exam
JEE Main
Year
2018
Date
15 April
Shift
Evening Shift
Mode
Online
Subject
Mathematics
Chapter
Matrices

JEE Main 2018 · 15 April · Evening Shift · Online · Mathematics · Matrices

Suppose A is any 3 × 3 non-singular matrix and (A-3 I)(A-5 I)=O ,...

Suppose AA is any 3×33 \times 3 non-singular matrix and (A−3I)(A−5I)=O(A-3 I)(A-5 I)=O, where I=I3I=I_3 and O=O3O=O_3. If αA+\alpha A+ βA−1=4I\beta A^{-1}=4 I, then α+β\alpha+\beta is equal to

Options

  1. A8
  2. B12
  3. C13
  4. D7

Correct answer

A. 8

Step-by-step solution

We have (A−3I)(A−5I)=O⇒A2−8A+15I=O\begin{aligned}&(A-3 I)(A-5 I)=O \\ &\Rightarrow A²-8 A+15 I=O \end{aligned}Multiplying both sides by A−1A^{-1}, we get; A−1A\cdotA−8A−1A+15A−1I=A−1O⇒A−8I+15A−1=OA+15A−1=8IA2+15A−12=4I∴α+β=12+152=162=8\begin{aligned}A^{-1} A \cdotA-8 A^{-1} A+15 A^{-1} I=A^{-1} O \\ \Rightarrow & A-8 I+15 A^{-1}=O \\ & A+15 A^{-1}=8 I \\ & \frac{A}{2}+\frac{15 A^{-1}}{2}=4 I \\ \therefore & \alpha+\beta=\frac{1}{2}+\frac{15}{2}=\frac{16}{2}=8 \end{aligned}
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