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JEE Advanced 2017 Paper 1 · Mathematics · Matrices and Determinants · medium

Let M=beginpmatrix2&1&1\\1&2&1\\1&1&2endpmatrix and let adjM denote its adjoint.

One or more options correct

Let M=(211121112)M=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix} and let adj⁡M\operatorname{adj}M denote its adjoint. Which of the following is (are) correct?
  1. A. det⁡(adj⁡(adj⁡M))=256\det\big(\operatorname{adj}(\operatorname{adj}M)\big)=256
  2. B. tr⁡(M−1)=52\operatorname{tr}\big(M^{-1}\big)=\dfrac52
  3. C. The (1,2)(1,2) entry of M5M^{5} is 341341
  4. D. M−1=14(5I−J)M^{-1}=\dfrac14\left(5I-J\right), where JJ is the 3×33\times3 matrix with every entry 11
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Answer: A, C

Write M=I+JM=I+J with J2=3JJ^2=3J. Then det⁡M=4\det M=4 (eigenvalues 4,1,14,1,1). (A) For an n×nn\times n matrix, det⁡(adj⁡(adj⁡M))=(det⁡M)(n−1)2=44=256\det(\operatorname{adj}(\operatorname{adj}M))=(\det M)^{(n-1)^2}=4^4=256; correct. (B) The eigenvalues of M−1M^{-1} are 14,1,1\tfrac14,1,1, so the trace is 94\tfrac94, not 52\tfrac52; incorrect. (C) By induction Mn=I+4n−13JM^n=I+\frac{4^n-1}{3}J, so the off-diagonal entry for n=5n=5 is 10233=341\frac{1023}{3}=341; correct. (D) Using J2=3JJ^2=3J, M−1=I−14J=14(4I−J)M^{-1}=I-\tfrac14J=\tfrac14(4I-J); check (I+J)(I−J4)=I+34J−34J=I(I+J)(I-\tfrac J4)=I+\tfrac34J-\tfrac34J=I. The stated 5I−J5I-J is wrong; incorrect. Answer: A and C. The shortcut is that M=I+JM=I+J has a rank-one part JJ satisfying J2=3JJ^2=3J, which makes both powers and the inverse easy to write down without any direct multiplication.
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Original question written in the JEE Advanced 2017 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake