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Let P = [(3, -1, -2), (2, 0, α), (3, - 5, 0)], where α ∈ R. Suppose Q = [qij] is a matrix such that PQ = kI, where k ∈R, k ≠ 0 and I is the identity matrix of order 3. If q23 = -k/8 = - and det(Q) = k2/2, then

(A) α = 0, k = 8

(B) 4α - k + 8 = 0

(C) det(P adj (Q)) = 29

(D) det(Q adj (P)) = 213

1 Answer

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Answer is (B) (C)

It is given that

Solving Eqs. (1) and (2), we get

−4 = −k ⇒ k = 4

Using the value of k in Eq. (1), we get

6α + 10 = 4 ⇒ α = −1

That is, α  = −1 and k = 4.

Hence, option (A) is incorrect.

The values of α  = −1 and k = 4 satisfy the equation given in option (B).

Hence, option (B) is correct.

Now, det Q = k2/2 =  8

Therefore, det(P adj Q) = (det P)det(adj Q) = (2 × 4)(detQ)2

= 8 × 82 = 23 × 26 = 29

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