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JEE Main 2019 · 12 Apr · Shift I

JEE Main 2019 Mathematics question on Matrices and Determinants

If AA is a symmetric matrix and BB is a skew-symmetric matrix such that A+B=[235−1]A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}, then ABAB is equal to
  1. A.
    [−4−2−14]\begin{bmatrix} -4 & -2 \\ -1 & 4 \end{bmatrix}
  2. B.
    [4−2−1−4]\begin{bmatrix} 4 & -2 \\ -1 & -4 \end{bmatrix}
  3. C.
    [4−21−4]\begin{bmatrix} 4 & -2 \\ 1 & -4 \end{bmatrix}
  4. D.
    [−4214]\begin{bmatrix} -4 & 2 \\ 1 & 4 \end{bmatrix}
Show answer and solution

Correct option: B

Solution

For any matrix M, its symmetric part is (M + Mᵀ)/2 and skew-symmetric part is (M − Mᵀ)/2.

Let M = A + B = [[2,3],[5,−1]]. Then Mᵀ = [[2,5],[3,−1]].

A = (M + Mᵀ)/2 = [[2,4],[4,−1]], B = (M − Mᵀ)/2 = [[0,−1],[1,0]].

AB = [[2,4],[4,−1]] · [[0,−1],[1,0]] = [[0+4, −2+0],[0−1, −4+0]] = [[4,−2],[−1,−4]].

[JEE Main 2019 · 12 Apr · Shift I]

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