Let λ and α be real. Then the number of integral values λ for which the system of linear equationsλx+(sin⁡α)y+(cos⁡α)z=0;x+(cos⁡α)y+(sin⁡α)z=0;−x+(sin⁡α)y−(cos⁡α)z=0has non-trivial solutions is

The given system has a non-trivial solution ifλsin⁡αcos⁡α1cos⁡αsin⁡α−1sin⁡α−cos⁡α=0By expanding the determinant along first column, we getλ=sin⁡2α+cos⁡2αWe know that−2≤sin⁡2α+cos⁡2α≤2∴ −2≤λ≤2Hence, integral values of λ are -1, 0 and 1.