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# If a matrix A is both symmetric and skew-symmetric, then (a) A is a diagonal matrix (b) A is a zero matrix (c) A is a scalar matrix (d) A is a square matrix

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Solution

## (b) A is a zero matrix Let $A=\left[{a}_{ij}\right]$ 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 ${a}_{ij}=-{a}_{ji}$ for all i, j $⇒{a}_{ji}=-{a}_{ij}$ for all i,j ...(2) From eqs. (1) and (2), we have ${a}_{ij}=-{a}_{ij}\phantom{\rule{0ex}{0ex}}⇒{a}_{ij}+{a}_{ij}=0\phantom{\rule{0ex}{0ex}}⇒2{a}_{ij}=0\phantom{\rule{0ex}{0ex}}⇒{a}_{ij}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\therefore A=\left[{a}_{ij}\right]\mathrm{is}\mathrm{a}\mathrm{zero}\mathrm{matrix}\mathrm{or}\mathrm{null}\mathrm{matrix}.\phantom{\rule{0ex}{0ex}}$

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