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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 think of is $A^T=-A$ for skew-symmetric matrix, and the other is $\operatorname{adj} A=(\operatorname{cofactor}A)^T$.

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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 $n$.

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  • $\begingroup$ How do we formulate adj(-A)=(-1)^n-1adjA, because i saw another one is like A^T=-A then det A=(-1)^ndetA $\endgroup$
    – Abec
    Commented Oct 15, 2018 at 23:43
  • $\begingroup$ 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. $\endgroup$
    – user593746
    Commented Oct 16, 2018 at 13:18

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