Tamil Nadu Board of Secondary EducationHSC Science Class 11th

If A = [42-1x] and such that (A – 2I)(A – 3I) = 0, find the value of x - Mathematics

Sum

If A = [(4, 2),(-1, x)] and such that (A – 2I)(A – 3I) = 0, find the value of x

Solution

A = [(4, 2),(-1, x)]

A – 2I = [(4, 2),(-1, x)] - 2[(1, 0),(0, 1)]

= [(4, 2),(-1, x)] - [(2, 0),(0, 2)]

= [(4 - 2, 2 - 0),(-1 - 0, x - 2)]

A – 2I = [(2, 2),(-1, x - 2)]

A – 3I = [(4, 2),(-1, x)] - 3[(1, 0),(0, 1)]

= [(4, 2),(-1, x)] - 3[(3, 0),(0, 3)]

= [(4 - 3, 2 - 0),(-1 - 0, x - 3)]

A – 3I = [(1, 2),(-1, x - 3)]

(A – 2I)(A – 3I) = [(2, 2),(-1, x  2)] [(1, 2),(-1, x - 3)]

= [(2 -2, 4 + 2(x - 3)),(-1 - (x - 2), -2 + (x - 2)(x - 3))]

= [(0, 4 + 2x- 6),(-1 - x + 2, -2 + x^2 - 3x - 2x + 6)]

(A – 2I)(A – 3I) = [(0, 2x - 2), (-x + 1, x^2 - 5x + 4)]

Given (A – 2I)(A – 3I) = 0

∴ [(0, 2x - 2), (-x + 1, x^2 - 5x + 4)] = [(0, 0),(0, 0)]

Equation the corresponding entries

2x – 2 = 0

⇒ x = 1

x2 – 5x + 4 = 0  ......(1)

Put x = 1 in equation (1)

12 – 5 × 1 + 4 = 5 – 5 = 0

∴ The  reqired value of x is x = 1

Concept: Matrices
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Chapter 7: Matrices and Determinants - Exercise 7.1 [Page 18]

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Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 7 Matrices and Determinants
Exercise 7.1 | Q 7 | Page 18
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