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JEE Advanced 2022 Paper 2 · Mathematics · Matrices and Determinants · medium

Let A=beginpmatrix1&2\\2&1endpmatrix and let I be the 2×2 identity matrix.

One or more options correct

Let A=(1221)A=\begin{pmatrix}1&2\\2&1\end{pmatrix} and let II be the 2×22\times2 identity matrix. Which of the following statements is/are correct?
  1. A. A2−2A−3I=OA^{2}-2A-3I=O
  2. B. A−1=2I−A3A^{-1}=\dfrac{2I-A}{3}
  3. C. trace⁡(A6)=728\operatorname{trace}(A^{6})=728
  4. D. For every positive integer nn, all entries of AnA^{n} are positive
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Answer: A, D

The characteristic polynomial is λ2−2λ−3\lambda^2-2\lambda-3, so by Cayley-Hamilton A2−2A−3I=OA^2-2A-3I=O; A is true. Rearranging, A(A−2I)=3IA(A-2I)=3I, hence A−1=A−2I3=(−1/32/32/3−1/3)A^{-1}=\frac{A-2I}{3}=\begin{pmatrix}-1/3&2/3\\2/3&-1/3\end{pmatrix}, which is the negative of the given expression; B is false. The eigenvalues are 33 and −1-1, so trace⁡(A6)=36+(−1)6=730\operatorname{trace}(A^6)=3^6+(-1)^6=730, not 728728; C is false. By diagonalisation An=12(3n+(−1)n3n−(−1)n3n−(−1)n3n+(−1)n)A^n=\frac12\begin{pmatrix}3^n+(-1)^n&3^n-(-1)^n\\3^n-(-1)^n&3^n+(-1)^n\end{pmatrix}, and 3n>13^n>1 for all n≥1n\ge1, so every entry is positive; D is true.
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