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Which one of the following is a fallacy?

$\begin {array} {1 1} (A)\;\text{The elements on the main diagonal of a symmetric matrix are all zero.} \\ (B)\;\text{The elements on the main diagonal of a skew-symmetric matrix are all zero.} \\ (C)\;\text{For any square matrix $ A, \large\frac{1}{2}(A+A’)$  is symmetric}\\ (D)\;\text{For any square matrix  $A, \large\frac{1}{2}(A-A’)$  is skew-symmetric.} \end {array}$

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Ans : (A)
Let $A=[aij]$ be a skew-symmetric matrix.
Then, $aij = - aij$ for all $I,j$
$aij = - aij$ for all $i=j$
$2\: aij=0$
$aij=0$ for all $i$
Now, let A be any square matrix, then
[since $ (A+B)’=A’+B’ ; (A’)’=A]$
$\large\frac{1}{2}(A+A’)$ is symmetric matrix.
Also, $ \large\frac{1}{2}(A-A’’=\large\frac{1}{2}[A’-(A’)’]=\large\frac{1}{2}(A’-A)= -\large\frac{1}{2}(A-A’)$
$ \large\frac{1}{2}(A-A’)$ is skew-symmetric matrix.
Hence, A is correct.


answered Jan 23, 2014 by thanvigandhi_1
edited Mar 21, 2014 by thanvigandhi_1

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