 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/18/b/solution

A square homogeneous linear system has a nonzero solution exactly when its determinant vanishes. Expanding along the second row gives

det A = 3 [(18 - λ) + 2 λ (1 - λ)] - 3 [(9 - λ) + λ (2 - λ)] = 3 (9 - λ^{2}) .

Consequently

λ = 3 or λ = - 3.

For an explicit check, the null space contains

(3, - 1, 1)^{T}

λ = 3

and

(3, 1, - 1)^{T}

λ = - 3

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