Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof resit / 4 / d

True. Transposing the inverse identities shows $(X^{-1}{)}^T=(X^T{)}^{-1}$. If $X^T=X$, this becomes $(X^{-1}{)}^T=X^{-1}$, exactly the symmetric matrix condition. The reasoning uses the matrix inverse and transpose identities, without assuming that different matrix factors commute.

True. Using the same transpose identity and $X^T=-X$ givesThe final equality follows by multiplying $-X$ and $-X^{-1}$ in either order and obtaining the identity. Thus the inverse is a skew-symmetric matrix as well. The invertibility hypothesis is retained throughout.