devarshi-dt-logo

Question:

Let k be a positive real number and let A = ⎡⎢⎢⎣2k 2√k 2√k√k k√k √k2k⎤⎥⎥⎦ and B = ⎡⎢⎢⎣0 2k √k k 0 2√k - √k √k 0⎤⎥⎥⎦. If det(adjA) + det(adjB) = 106, then [k] is equal to [Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].

4

5

6

3

Solution:

|A| = (2k + 1)^3, |B| = 0 (Since B is a skew-symmetric matrix of order 3) ⇒ det(adjA) = |A|^(n-1) = ((2k + 1)^3)^2 = 106 ⇒ 2k + 1 = ∛106 ⇒ 2k = ∛106 -1. Since ∛106 ≈ 4.72, 2k ≈ 3.72, k ≈ 1.86. Therefore, [k] = 4.