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(D) Given that $B$ is a skew-symmetric matrix,we have $B^{\prime} = -B$. To check if $A^{\prime} B A$ is symmetric or skew-symmetric,we take its trans

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Skew-symmetric matrix

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(D) Given that $B$ is a skew-symmetric matrix,we have $B^{\prime} = -B$. To check if $A^{\prime} B A$ is symmetric or skew-symmetric,we take its transpose: $(A^{\prime} B A)^{\prime} = A^{\prime} B^{\prime} (A^{\prime})^{\prime}$ Using the property $(XYZ)^{\prime} = Z^{\prime} Y^{\prime} X^{\prime}$,we get: $(A^{\prime} B A)^{\prime} = A^{\prime} B^{\prime} A$ Since $B^{\prime} = -B$,we substitute this into the expression: $(A^{\prime} B A)^{\prime} = A^{\prime} (-B) A = -(A^{\prime} B A)$ Since the transpose of the matrix $A^{\prime} B A$ is equal to its negative,$A^{\prime} B A$ is a skew-symmetric matrix.

If $A$ and $B$ are square matrices of the same order and $B$ is a skew-symmetric matrix,then $A^{\prime} B A$ is

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