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JEE Main 2013

JEE Main 2013 Mathematics question on Matrices and Determinants

If P=[1α3133244]P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix} is the adjoint of a 3×33 \times 3 matrix AA and ∣A∣=4|A| = 4, then α\alpha is equal to
  1. A.
    4
  2. B.
    11
  3. C.
    5
  4. D.
    0
Show answer and solution

Correct option: B

Solution

For a 3×3 matrix A, the key relation is adj(A) · A = |A| · I, which implies |adj(A)| = |A|^(n-1) = 42 = 16.

So det(P) = 16. Computing det(P) by expanding:

det(P) = 1·(3·4 - 3·4) - α·(1·4 - 3·2) + 3·(1·4 - 3·2)
= 1·(12-12) - α·(4-6) + 3·(4-6)
= 0 - α·(-2) + 3·(-2)
= 2α - 6

Setting 2α - 6 = 16 gives 2α = 22, so α = 11.

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