 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/18/d/solution

det A_{α} = 10 (α - \frac{1}{2})^{2} + \frac{11}{2} > 0

for every real

α

. A square matrix is invertible exactly when its determinant is nonzero, so $A_{α}$ is nonsingular for every real $α$ .

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