Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 6D / c / Solution

True. Use the odd-dimensional orthogonal determinant splitting:Scalar matrices commute with every matrix, so $\Phi (AB)=\Phi (A)\Phi (B)$. Its output is orthogonal andEvery $B\in SO(3)$ satisfies $\Phi (B)=B$, proving surjectivity. The equation $\Phi (A)=I$ forces $A=(\det A)I$, whenceIn odd dimension $-I$ has determinant $-1$, which is the sign needed in this construction.