Question

If A^T denotes the transpose of the matrix A = `[(0, 0, a),(0, b, c),(d, e, f)]`, where a, b, c, d, e and f are integers such that abd ≠ 0, then the number of such matrices for which A^–1 = A^T is ______.

Options

• 2(3!)

• 3(2!)

• 2³

• 3²

Answer 1

If A^T denotes the transpose of the matrix A = `[(0, 0, a),(0, b, c),(d, e, f)]`, where a, b, c, d, e and f are integers such that abd ≠ 0, then the number of such matrices for which A^–1 = A^T is `underlinebb(2^3)`.

Explanation:

A = `[(0, 0, a),(0, b, c),(d, e, f)]`, |A| = – abd ≠ 0

c₁₁ = + (bf – ce), c₁₂ = – (– cd) = cd, c₁₃ = + (– bd) = – bd

c₂₁ = – (– ea) = ae, c₂₂ = + (– ad) = –ad, c₂₃ = – (0) = 0

c₃₁ = + (–ab) = – ab, c₃₂ = – (0) = 0, c₃₃ = 0

Adj A = `[((bf - ce), ae, -ab),(cd, -ad, 0),(-bd, 0, 0)]`

A^–1 = `1/|A|(adj A) = 1/(abd) [(bf - ce, ae, -ab),(cd, -ad, 0),(-bd, 0, 0)]`

A^T = `[(0, 0, d),(0, b, e),(a, c, f)]` Now A^–1 = A^T

`\implies 1/(-abd) [(bf - ce, ae, -ab),(cd, -ad, 0),(-bd, 0, 0)] = [(0, 0, d),(0, b, e),(a, c, f)]`

`\implies [(bf - ce, ae, -ab),(cd, -ad, 0),(-bd, 0, 0)] =  [(0, 0, -abd^2),(0, -ab^2d, -abde),(-a^2bd, -abcd, -abdf)]`

∴ bf – ce = ae = cd = 0  ...(i)

abd² = ab, ab²d = ad, a²bd = bd  ...(ii)

abde = abcd = abdf = 0 ....(iii)

From (ii),

(abd²). (ab²d). (a²bd) = ab. ad. bd

`\implies` (abd)⁴ – (abd)² = 0

`\implies` (abd)² [(abd)² – 1] = 0 ∵ abd ≠ 0, ∴ abd = ±1  ....(iv)

From (iii) and (iv), e = c = f = 0  ...(v) 

From (i) and (v), bf = ae = cd = 0  ...(vi)

From (iv), (v) and (vi), it is clear that a, b, d can be any non-zero integer such that abd = ± 1

But it is only possible, if a = b = d = ± 1

Hence, there are 2 choices for each a, b and d.

Therefore, there are 2 × 2 × 2 choices for a, b and d.

Hence number of required matrices = 2 × 2 × 2 = (2)³