Let A α = cos ⁡ α 0 sin ⁡ α 0 1 0 sin ⁡ α 0 cos ⁡ α and A x y z = 0 1 0 . If the system of equations has infinite solutions and sum of all the possible value of α in 0,2 π is k π , then the value of k is equal to

Step-by-step solution

All the three equations obtained are cos ⁡ α x + sin ⁡ α z = 0 y = 1 sin ⁡ α x + cos ⁡ α z = 0 So, the equations cos ⁡ α x + sin ⁡ α z = 0 and sin ⁡ α x + cos ⁡ α z = 0 has infinite solutions Then, ∆ = cos ⁡ α sin ⁡ α sin ⁡ α cos ⁡ α = 0 ⇒ cos 2 ⁡ α - sin 2 ⁡ α = 0 ⇒ cos ⁡ 2 α = 0 ⇒ α = π 4 , 3 π 4 , 5 π 4 , 7