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प्रश्न
Without expanding evaluate the following determinant:
`|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`
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उत्तर
Let D = `|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`
`C_1 = [(2),(3),(5)], C_2 = [(7),(8),(9)], C_3 = [(65),(75),(86)]`
C3 = aC1 + bC2
65 = 2a + 7b
75 = 3a + 8b
86 = 5a + 9b
2a + 7b = 65 ...(i)
3a + 8b = 75 ...(ii)
Multiply (i) by 3:
6a + 21b = 195 ...(iii)
Multiply (ii) by 2:
6a + 16b = 150 ...(iv)
(6a + 21b) − (6a + 16b) = 195 − 150
5b = 45 ⇒ b = 9
2a + 7(9) = 65 ⇒ 2a = 65 − 63 = 2 ⇒ a = 1
C3 = 1 ⋅ C1 + 9 ⋅ C2
If one column is a linear combination of others, the determinant is zero.
= 0
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