Question

Without expanding evaluate the following determinant:

`|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`

Answer 1

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​ + bC_2​

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