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JEE Main 2017

JEE Main 2017 Mathematics question on Matrices and Determinants

If A=[2−3−41]A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}, then adj(3A2+12A)\text{adj}(3A^2 + 12A) is equal to
  1. A.
    [72−84−6351]\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}
  2. B.
    [51638472]\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}
  3. C.
    [51846372]\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}
  4. D.
    [72−63−8451]\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}
Show answer and solution

Correct option: B

Solution

First, compute A² and 12A.

A² = [[2,-3],[-4,1]]² = [[2·2+(-3)(-4), 2(-3)+(-3)(1)],[(-4)(2)+(1)(-4), (-4)(-3)+(1)(1)]] = [[16,-9],[-12,13]]

3A² = [[48,-27],[-36,39]]

12A = [[24,-36],[-48,12]]

3A² + 12A = [[72,-63],[-84,51]]

For a 2×2 matrix M = [[a,b],[c,d]], adj(M) = [[d,-b],[-c,a]].

So adj([[72,-63],[-84,51]]) = [[51,63],[84,72]].

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