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JEE Advanced 2023 · Paper 2 · one or more correct

JEE Advanced 2023 Mathematics question on Matrices and Determinants

Let M=(aij)M = (a_{ij}), i,j∈{1,2,3}i, j \in \{1, 2, 3\}, be the 3×33 \times 3 matrix such that aij=1a_{ij} = 1 if j+1j + 1 is divisible by ii, otherwise aij=0a_{ij} = 0. Then which of the following statements is(are) true?
  1. A.
    MM is invertible
  2. B.
    There exists a nonzero column matrix (a1a2a3)\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} such that M(a1a2a3)=(−a1−a2−a3)M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix}
  3. C.
    The set {X∈R3:MX=0}≠{0}\{X \in \mathbb{R}^3 : MX = \mathbf{0}\} \neq \{\mathbf{0}\}, where 0=(000)\mathbf{0} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
  4. D.
    The matrix (M−2I)(M - 2I) is invertible, where II is the 3×33 \times 3 identity matrix
Show answer

Correct options: B, C

[JEE Advanced 2023 · Paper 2]

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