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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 3

The three eigenvalues are
$$
1,\qquad e^{i\pi/4}=\frac{1+i}{\sqrt2},
\qquad e^{3i\pi/4}=\frac{-1+i}{\sqrt2}.
$$
Thus $N(U)$ is the filled triangle with vertices
$$
(1,0),\qquad(2^{-1/2},2^{-1/2}),
\qquad(-2^{-1/2},2^{-1/2})
$$
in the complex plane. Its upper edge is the horizontal chord joining the last two vertices, and the other two edges join those points to $1$. In particular, the origin lies outside this triangle.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / d / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / d / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / d / Solution

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