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 ______.

पर्याय

• 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 17.

Explanation:

We are given PQ = kI₃, so:

`Q = kP^(-1) = k/(|P|) * adj(P)`

Given:

`q_23 = -k/8 => (adj(P))_23 = -(|P|)/8`

`|Q| = (k^2)/2 = (k^3)/(|P|) => |P| = 2k`

Compute the cofactor (adj(P))₂₃ = −(3α + 4), so:

`-(3alpha + 4) = -(|P|)/8`

⇒ 3α + 4 = `k/4`

⇒ α = `(k - 16)/12`

`alpha^2 + k^2 = ((k - 16)/12)^2 + k^2`

Try k = 4:

`alpha = (4 - 16)/12 = -1`

α² + k² = 1 + 16 = 17