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

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- Solution d

Solution

Completing the square in the determinant gives

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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