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Question
If the matrix A is both symmetric and skew symmetric, then ______.
Options
A is a diagonal matrix
A is a zero matrix
A is a square matrix
None of these
Solution
If the matrix A is both symmetric and skew symmetric, then A is a zero matrix.
Explanation:
In symmetric matrices, aij = aji …(1)
In skew symmetric matrices, aij = -aji …(2)
Symmetric and skew-symmetric matrices must have both properties (1) and (2). Combining them,
2aij = aij - aji = 0
⇒ aij = 0
aij = aji 0
∴ The square matrix will be a zero matrix.
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