Let , where a ∈ 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 = and det (Q) = , then

Engineering

Mathematics

Inverse of a Matrix

Question

Let $P = [\begin{matrix} 3 & - 1 & - 2 \\ 2 & 0 & α \\ 3 & - 5 & 0 \end{matrix}]$ , where a ∈ R. Suppose Q = [q_ij] is a matrix such that PQ = kI, where k ∈ R, k ≠ 0 and I is the identity matrix of order 3. If q₂₃ = $\frac{- k}{8}$ and det (Q) = $\frac{k^{2}}{2}$ , then

A

α = 0, k = 8

B

det (P adj (Q)) = 2⁹

C

det (Q adj (P)) = 2¹³

D

4α – k + 8 = 0

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Q = P^–1k

| Q | = $\frac{k^{3}}{| P |} = \frac{k^{2}}{2}$

| P | = 2k

$Q = \frac{k (adj . P)}{| P |} = \frac{k}{| P |} adj . (P) = \frac{- k}{| P |} (3 α + 4) = \frac{- k}{8}$

⇒ 24α + 32 = | P |

| P | = 12α + 20

24α + 32 = 12α + 20 ⇒ α = – 1

| P | = 8, k = 4

(B) 4α – k + 8 = 0

(C) | P adj. (Q) | = | P | | Q |² = 8 · 8² = 2⁹

(D) | Q (adj. P) | = | Q | | adj. P | = | Q | · | P |² = 8³ = 2⁹.