If a matrix A is both symmetric and skew-symmetric, then

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A is a zero matrix A = \left[ a_{ij} ight] be a matrix which is both symmetric and skew-symmetric. If A = \left[ a_{ij} ight] is a symmetric matrix, then a_{ij} = a_{ji} for all i, j ............(1) If A = \left[ a_{ij} ight] 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 	herefore A = \left[ a_{ij} ight] 	ext{is a zero matrix or null matrix} .

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