If $$A$$ is a skew-symmetric $$n\times n$$ matrix, verify that $$\operatorname{adj} A$$ is symmetric or skew-symmetric according to whether $$n$$ is odd or even.
Things I can thought of is $$A^T=-A$$ for skew-symmetric matrix, and the other is $$\operatorname{adj} A=(\operatorname{cofactor}A)^T$$.
Here are some hints. Show that $$\operatorname{adj}(A^T)=(\operatorname{adj} A)^T$$ for every square matrix $$A$$. Then, show that $$\operatorname{adj}(-A)=(-1)^{n-1}\operatorname{adj} A$$ for every square matrix $$A$$ of order $$n$$.
• What is the definition of $\operatorname{adj} A$? If you take a look at the definition carefully, your knowledge that $\det(-A)=(-1)^n\det A$ will become handy. – user593746 Oct 16 '18 at 13:18