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(B) Let $C = AB - BA$.Taking the transpose of $C$, we get $C^T = (AB - BA)^T$.Using the property $(X - Y)^T = X^T - Y^T$ and $(XY)^T = Y^T X^T$, we ha

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Skew-symmetric matrix

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(B) Let $C = AB - BA$.Taking the transpose of $C$, we get $C^T = (AB - BA)^T$.Using the property $(X - Y)^T = X^T - Y^T$ and $(XY)^T = Y^T X^T$, we have $C^T = (AB)^T - (BA)^T = B^T A^T - A^T B^T$.Since $A$ and $B$ are skew-symmetric, $A^T = -A$ and $B^T = -B$.Substituting these values, $C^T = (-B)(-A) - (-A)(-B) = BA - AB$.Factoring out a negative sign, $C^T = -(AB - BA) = -C$.Since $C^T = -C$, the matrix $AB - BA$ is a skew-symmetric matrix.

If $A$ and $B$ are skew-symmetric matrices of the same order, then $AB - BA$ is a . . . . . . .

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