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← JEE Advanced 2020 · Paper 1
MathematicsMultiple correct+4 / -2Question 44 of 54

JEE Advanced 2020 · Paper 1, Mathematics question 8

Let MM be a 3×33\times3 invertible matrix with real entries and let II denote the 3×33\times3 identity matrix. If M−1=adj⁡(adj⁡M)M^{-1} = \operatorname{adj}(\operatorname{adj} M), then which of the following statements is/are ALWAYS TRUE?

Choose one or more options
Answer and worked solution

Official answer: B, C, D

Simplify. For a 3×33\times3 matrix, adj⁡(adj⁡M)=(det⁡M)M\operatorname{adj}(\operatorname{adj}M)=(\det M)M. So
M−1=(det⁡M)M  ⟹  M2=1det⁡MI.M^{-1}=(\det M)M\implies M^2=\frac1{\det M}I.

Determinant. Taking determinants, (det⁡M)2=(det⁡M)−3(\det M)^2=(\det M)^{-3}, so (det⁡M)5=1(\det M)^5=1 and det⁡M=1\det M=1 (it is real). (B) is true.

Consequences.

  • M2=IM^2=I, so (C) is true.
  • adj⁡M=(det⁡M)M−1=M−1\operatorname{adj}M=(\det M)M^{-1}=M^{-1}, so (adj⁡M)2=M−2=I(\operatorname{adj}M)^2=M^{-2}=I. (D) is true.

(A) M=diag⁡(1,−1,−1)M=\operatorname{diag}(1,-1,-1) satisfies everything but is not II, so (A) is not always true.

(B), (C), (D)

Topics:Adjoint and inverseDeterminants and properties
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