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# If the matrix A is both symmetric and skew symmetric, then

$(a)\;A\;is\;a\;diagonal\;matrix\qquad(b)\;A\;is\;a\;zero\;matrix\qquad(c)\;A\;is\;a\;square\;matrix\qquad(d)\;None\;of\;these$

Toolbox:
• A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j.
• A square matrix A=[a$_{ij}$] is said to be skew symmetric if A'=-A that is $[a_{ij}]= -[a_{ji}]$ for all possible value of i and j.
Step 1: If a matrix is both symmetric and skew symmetric matrix ,then
A is symmetric matrix
$\Rightarrow a_ij=aji$
A is a skew symmetric matrix
$\Rightarrow a_{ij}=-a_{ji}$
Step 2: If $a_{ij}=a_{ji}=-a_{ji}$
$\Rightarrow a_{ij}=0$
Hence A is a zero matrix.
so (B) is the correct answer.
edited Mar 20, 2013