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Question
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
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