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Question
If a matrix A is both symmetric and skew-symmetric, then
Options
A is a diagonal matrix
A is a zero matrix
A is a scalar matrix
A is a square matrix
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Solution
A is a zero matrix
\[A = \left[ a_{ij} \right]\] be a matrix which is both symmetric and skew-symmetric.
If \[A = \left[ a_{ij} \right]\] is a symmetric matrix, then
\[a_{ij} = a_{ji}\] for all i, j ............(1)
If \[A = \left[ a_{ij} \right]\] is a skew-symmetric matrix, then
\[a_{ij} = - a_{ji}\]
\[\Rightarrow a_{ji} = - a_{ij}\] for all i,j ............(2)
From eqs. (1) and (2), we have
\[a_{ij} = - a_{ij} \]
\[ \Rightarrow a_{ij} + a_{ij} = 0 \]
\[ \Rightarrow 2 a_{ij} = 0 \]
\[ \Rightarrow a_{ij} = 0 \]
\[ \therefore A = \left[ a_{ij} \right] \text{is a zero matrix or null matrix} . \]
\[\]
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