Tamil Nadu Board of Secondary EducationHSC Arts Class 12

If A = 19[-8144471-84], prove that AATA-1=AT - Mathematics

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Sum

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

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Solution

R.H.S : AT = `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) ⇒ AT = A–1 

Concept: Inverse of a Non-singular Square Matrix
  Is there an error in this question or solution?
Chapter 1: Applications of Matrices and Determinants - Exercise 1.1 [Page 15]

APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 12th Mathematics Volume 1 and 2 Answers Guide
Chapter 1 Applications of Matrices and Determinants
Exercise 1.1 | Q 5 | Page 15

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