Question

Let P = `[(1, 0, 0),(3, 1, 0),(9, 3, 1)]` and Q = [q_ij] be two 3 × 3 martices such that Q – P⁵ = I₃. Then `(q_21 + q_31)/q_32` is equal to ______.

विकल्प

• 10

• 135

• 15

• 9

Answer 1

Let P = `[(1, 0, 0),(3, 1, 0),(9, 3, 1)]` and Q = [q_ij] be two 3 × 3 martices such that Q – P⁵ = I₃. Then `(q_21 + q_31)/q_32` is equal to 10.

Explanation:

P² = `[(1, 0, 0),(3, 1, 0),(9, 3, 1)][(1, 0, 0),(3, 1, 0),(9, 3, 1)] = [(1, 0, 0),(6, 1, 0),(27, 6, 1)]` 

`\implies` P⁴ = `[(1, 0, 0),(6, 1, 0),(27, 6, 1)][(1, 0, 0),(6, 1, 0),(27, 6, 1)] = [(1, 0, 0),(12, 1, 0),(90, 12, 1)]`

`\implies` P⁵ = `[(1, 0, 0),(12, 1, 0),(90, 12, 1)][(1, 0, 0),(3, 1, 0),(9, 3, 1)] = [(1, 0, 0),(15, 1, 0),(135, 15, 1)]`

∵ Q – P⁵ = I₃ 

∴ Q = I₃ + P⁵ = ` [(2, 0, 0),(15, 2, 0),(135, 15, 2)]`

`(q_21 + q_31)/q_32 = (15 + 135)/15` = 10