Question

Let P = `[(3, -1, -2),(2, 0, alpha),(3, -5, 0)]`, where α ∈ R. Suppose Q = [q_ij] is a matrix satisfying PQ = kI₃ for some non-zero k ∈ R. If q₂₃ = `-k/8` and |Q| = `k^2/2`, then α² + k² is equal to ______.

Options

• 14

• 15

• 16

• 17

Answer 1

Let P = `[(3, -1, -2),(2, 0, α),(3, -5, 0)]`, where α ∈ R. Suppose Q = [q_ij] is a matrix satisfying PQ = kI₃ for some non-zero k ∈ R. If q₂₃ = `-k/8` and |Q| = `k^2/2`, then α² + k² is equal to 15.

Explanation:

P = `[(3, -1, -2),(2, 0, α),(3, -5, 0)]`, α ∈ R

PQ = kI₃ for some non-zero k ∈ R

Where Q = [q_ij]

q₂₃ = `(-k)/8` and |Q| = `k^2/2`

PQ = kI

Multiply both sides by P^–1

P^–1PQ = kP^–1I

P^–1P = I, P^–1I = P^–1

∴ Q = kP^–1

⇒ `1/kQ` = P^–1

Q = `k1/|p|` (adj(P)).I

q₂₃ = `(-k)/8` ⇒ 2^nd row, 3^rd column element in P^–1 is `(-1)/8`

Also |P| = 3(0 + 5α)+ 1(0 – 3α)2(–10)

= 15α – 3α + 20

= 12α + 20

⇒ `1/(20 + 12α)`(cofactor of P₃₂) = `(-1)/8`

= `1/(20 + 12α)(-(3α + 4)) = (-1)/8`

⇒ 2(3α + 4) = 5 + 3α

⇒ α = –1

and `|1/kQ| = |P^-1| = 1/(20 + 12α) = 1/8`  ...`{∵ Q = k^2/2}`

`1/k^3|Q| = 1/8` ⇒ `k^2/(2k^3)` ⇒ k = 4

Now, α² + k²

= (–1)² + (4)²

= 15