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Mathematics · Matrices

JEE Advanced 2020 — Paper 1 — Question 36

Let M be a 3×33 \times 3 invertible matrix with real entries and let I denote the 3×33 \times 3 identity matrix. If M−1=adj⁡(adj⁡M)\mathrm{M}^{-1}=\operatorname{adj}(\operatorname{adj} \mathrm{M}), then which of the following statements is/are ALWAYS TRUE?

  1. Option A:

    M=I\mathrm{M}=\mathrm{I}

  2. Option B:

    det⁡M=1\operatorname{det} \mathrm{M}=1

    Correct
  3. Option C:

    M2=I\mathrm{M}^{2}=\mathrm{I}

    Correct
  4. Option D:

    (adj⁡M)2=I(\operatorname{adj} M)^{2}=I

    Correct

Answer: B, C, D

Step-by-step solution

M−1=adj M∣M∣M^{-1} = \frac{\text{adj } M}{|M|} ⇒adj M=∣M∣M−1\Rightarrow \text{adj } M = |M| M^{-1} M−1=adj (M−1)∣M−1∣M^{-1} = \text{adj } (M^{-1}) |M^{-1}| =adj (adj M∣M∣)1∣M∣= \text{adj } \left( \frac{\text{adj } M}{|M|} \right) \frac{1}{|M|} M−1=1∣M∣adj(adj M∣M∣)M^{-1} = \frac{1}{|M|} \text{adj} \left( \frac{\text{adj } M}{|M|} \right) M adj M=∣M∣IM \text{ adj } M = |M| I replace M by adj M\text{replace } M \text{ by adj } M (adj M) adj (adj M)=∣adj M∣I=∣M∣n−1I(\text{adj } M) \text{ adj } (\text{adj } M) = |\text{adj } M| I = |M|^{n-1} I For n=3,(adj M) adj (adj M)=∣M∣2I\text{For } n=3, (\text{adj } M) \text{ adj } (\text{adj } M) = |M|^2 I Multiplying by M\text{Multiplying by } M M(adj M) adj (adj M)=∣M∣2MM (\text{adj } M) \text{ adj } (\text{adj } M) = |M|^2 M ∣M∣I adj (adj M)=∣M∣2M|M| I \text{ adj } (\text{adj } M) = |M|^2 M adj (adj M)=∣M∣M\text{adj } (\text{adj } M) = |M| M M−1=adj (M)M^{-1} = \text{adj } (M) =∣M∣M−1= |M| M^{-1} ⇒∣M−1∣=∣∣M∣M−1∣=∣M∣n∣M−1∣\Rightarrow |M^{-1}| = ||M| M^{-1}| = |M|^n |M^{-1}| 1∣M∣=∣M∣n1∣M∣\frac{1}{|M|} = |M|^n \frac{1}{|M|} 1=∣M∣n1 = |M|^n For n=3,1=∣M∣3⇒∣M∣=1.....(1)\text{For } n=3, 1 = |M|^3 \Rightarrow |M| = 1 \quad ..... (1) adj M∣M∣=∣M∣M−1\frac{\text{adj } M}{|M|} = |M| M^{-1} adj M=∣M∣2M−1\text{adj } M = |M|^2 M^{-1} adj M=∣M∣2adj M\text{adj } M = |M|^2 \text{adj } M adj M=M as ∣M∣=1\text{adj } M = M \text{ as } |M| = 1 ⇒M(adj M)=M2\Rightarrow M (\text{adj } M) = M^2 ⇒∣M∣I=M2\Rightarrow |M| I = M^2 ⇒M2=I\Rightarrow M^2 = I adj M=M as ∣M∣=1\text{adj } M = M \text{ as } |M| = 1 (adj M)2=I (as adj M=M)(\text{adj } M)^2 = I \text{ (as adj } M = M)

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2020
Paper
Paper 1
Subject
Mathematics
Chapter
Matrices
Topic
Adjoint of a Square Matrix
Let M be a 3 × 3 invertible matrix with real entries and let I denote… | JEE Advanced 2020 PYQ with Solution · DhiX AI