Question

If A = `1/9[(-8, 1, 4),(4, 4, 7),(1, -8, 4)]`, prove that `"A"^-1 = "A"^"T"`

Answer 1

R.H.S : A^T = `1/9[(-8, 4, 1),(1, 4, -8),(4, 7, 4)]`  ........(1)

L.H.S : If  A is martix of order n = 3

|A| = `(1/9)^3 [-8(16 + 56) - 1(16 - 7) + 4(- 32 - 4)]`

∵ |kA| = kⁿ |A|

= `1/729 [-8(72) - 1(9) + 4(- 36)]`

= `/729 (- 576 - 9 - 144)`

= `1/729 (- 729)`

= – 1 ≠ 0

∴ A^–1 exists.

adj A = `(1/9)^(3 - 1) [(+|(4, 7),(-8, 4)|, -|(4, 7),(1, 4)|, +|(4, 4),(1, -8)|),(-|(1, 4),(-8, 4)|, +|(-8, 4),(1, 4)|, -|(-8, 1),(1, -8)|),(+|(1, 4),(4, 7)|, -|(-8, 4),(4, 7)|, +|(-8, 1),(4, 4)|)]^"T"`

∵ `"adj" (lambda"A") = lambda^("n" - 1) ("adj A")`

= `1/81 [(+(16 + 56), -(16 - 7), +(-32 - 4)),(-(4 + 32), +(-32 - 4), -(64 - 1)),(+(7 - 16), -(-56 - 16),+(-32 - 4))]`

= `1/81 [(72, -9, -36),(-36, -36, -63),(-9, 72, -36)]^"T"`

adj A = `1/81 [(72, -36, -9),(-9, -36, 72),(-36, -63, -36)]`

= `1/81 xx 9[(8, 4, -1),(-1, -4, 8),(-4, -7, -4)]`

= `1/9 [(8, -4, -1),(-1, -4, 8),(-4, -7, -4)]`

A^–1 = `1/|"A"|` adj A

= `1/(-1) * 1/9 [(8, -4, -1),(-1, -4, 8),(-4, -7, -4)]`

A^–1 = `1/9 [(-8, 4, 1),(1, 4, -8),(4, 7, 4)]`  ........(2)

(1), (2) ⇒ A^T = A^–1