to show $(I_n+S)$ and $(I_n-S)$ are non-singular if $S$ is skew symmetric

Question

If $S$ be a real skew-symmetric matrix of order $n$ , prove that the matrices $(I_n+S)$ and $(I_n-S)$ are both non-singular.

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can anyone help me please to solve this problem.thanks for your help.

Answer 1

The product $(I_n + S)(I_n - S) = I_n - S^2 = I + S^T S$ which is positive definite and therefore invertible

Answer 2

I think muzzlator's answer is what the question setter had in mind. For an alternative proof, show that every real eigenvalue of a real skew symmetric matrix is zero. Hence $I+S$ and $I-S$ must be nonsingular, otherwise $1$ or $-1$ would be a real eigenvalue of $S$.