Consider a rotation T (x) = Ax- in R3. (That is, A is an orthogonal 3 × 3 matrix with determinant 1.) Show that T has a nonzero fixed point [i. e., a vector v- with T (v) -= v]. This result is known as Euler’s theorem, after the great Swiss mathematician Leonhard Euler (1707–1783). Hint: Consider the characteristic polyno- mial fA(λ). Pay attention to the intercepts with both axes. Use Theorem 7.1.4.