Paper 1, Section II, 6 A6 \mathrm{~A}

Vectors and Matrices | Part IA, 2011

The map Φ(x)=n×(x×n)+α(nx)n\Phi(\mathbf{x})=\mathbf{n} \times(\mathbf{x} \times \mathbf{n})+\alpha(\mathbf{n} \cdot \mathbf{x}) \mathbf{n} is defined for xR3\mathbf{x} \in \mathbb{R}^{3}, where n\mathbf{n} is a unit vector in R3\mathbb{R}^{3} and α\alpha is a constant.

(a) Find the inverse map Φ1\Phi^{-1}, when it exists, and determine the values of α\alpha for which it does.

(b) When Φ\Phi is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.

(c) Let y=Φ(x)\mathbf{y}=\Phi(\mathbf{x}). Find the components AijA_{i j} of the matrix AA such that yi=Aijxjy_{i}=A_{i j} x_{j}. When Φ\Phi is invertible, find the components of the matrix BB such that xi=Bijyjx_{i}=B_{i j} y_{j}.

(d) Now let AA be as defined in (c) for the case n=13(1,1,1)\mathbf{n}=\frac{1}{\sqrt{3}}(1,1,1), and let

C=13(221122212)C=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{array}\right)

By analysing a suitable determinant, for all values of α\alpha find all vectors x\mathbf{x} such that Ax=CxA \mathbf{x}=C \mathbf{x}. Explain your results by interpreting AA and CC geometrically.

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