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Question
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
Solution
Let A = [aij] be a matrix which is both symmetric and skew-symmetric.
Since A is a skew-symmetric matrix, so A′ = –A.
Thus for all i and j, we have aij = – aji ......(1)
Again, since A is a symmetric matrix, so A′ = A.
Thus, for all i and j, we have
aji = aij ......(2)
Therefore, from (1) and (2), we get
aij = – aij for all i and j
or
2aij = 0
i.e., aij = 0 for all i and j.
Hence A is a zero matrix.
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