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If a matrix AA is both symmetric and skew-symmetric, then AA will be the zero matrix.
Let's denote AA as the matrix:
A=[aij]A=[aij]
Symmetric Matrix: A matrix is symmetric if it is equal to its transpose. Mathematically, AT=AAT=A. In other words, for every ii and jj, aij=ajiaij=aji.
Skew-Symmetric Matrix: A matrix is skew-symmetric if its transpose is equal to the negative of itself. Mathematically, AT=−AAT=−A. In other words, for every ii and jj, aij=−ajiaij=−aji.
Combining these two conditions, we have aij=ajiaij=aji and aij=−ajiaij=−aji.
The only number that satisfies both conditions simultaneously is aij=0aij=0, because it's the only number that is equal to its negative.
Therefore, in this case, every element of matrix AA must be zero, making AA the zero matrix.
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