Chapter 2 - Matrices and Systems of Linear Equations - 2.6 The Inverse of a Square Matrix - Problems - Page 178: 33

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$A$ is an $n \times n$ invertible skew-symmetric matrix, we have $A^T=-A\\ (A^T)^{-1}=(-A)^{-1}$ Since $(-A)^{-1}(-A)=(-1)(-1)(A^{-1}A)=A^{-1}A=I_n$ and $(-A)(-A^{-1})=(-1)(-1)(AA^{-1})=I_n$ then $(-A)^{-1}=-A^{-1}\\ \rightarrow (A^T)^{-1}=-A^{-1}\\ \rightarrow (A^{-1})^T=-A^{-1}$ Hence, $A^{-1}$ is skew-symmetric.