(i) Suppose $A$ is a matrix that does not have $-1$ as an eigenvalue. Show that $A+I$ is non-singular. Further, show that $A$ commutes with $(A+I)^{-1}$.

(ii) A matrix $A$ is called skew-symmetric if $A^{T}=-A$. Show that a real skewsymmetric matrix does not have $-1$ as an eigenvalue.

(iii) Suppose $A$ is a real skew-symmetric matrix. Show that $U=(I-A)(I+A)^{-1}$ is orthogonal with determinant 1 .

(iv) Verify that every orthogonal matrix $U$ with determinant 1 which does not have $-1$ as an eigenvalue can be expressed as $(I-A)(I+A)^{-1}$ where $A$ is a real skew-symmetric matrix.