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A square matrix is called skew-symmetric if $A^T=-A$.
Prove that if $A$ and $B$ are skew-symmetric matrices, then $A+B$ is skew symmetric.

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We have

$$(A+B)^T = B^T + A^T = -B - A = -(B + A) = -(A + B).$$

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Let $A,B$ be $n\times n$ matrices skew-symmetric. Then $$ (A+B)^{\top}_{ij} = (A^{\top}_{ij} + B^{\top}_{ij}) = (-A_{ij} - B_{ij}) = -(A+B)_{ij} $$ for all $1 \leq i,j \leq n$, i.e. $(A+B)^{\top} = -(A+B).$

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