A) The elements on the main diagonal of a symmetric marix are all zero.
B) The elements on the main diagonal of a skew-symmetric matrix are all zero.
C) For any square matrix \[A,\frac{1}{2}(A+A')\] is symmetric.
D) For any square matrix \[A,\frac{1}{2}(A+A')\] is skew-symmetric.
Correct Answer: A
Solution :
Let \[A=[{{a}_{ij}}]\] be a skew-symmetric matrix. Then, \[{{a}_{ij}}=-{{a}_{ji}}\] for all i, j. \[\Rightarrow \]\[{{a}_{ij}}=-{{a}_{ij}}\]for all values of \[i=j\] \[\Rightarrow \]\[2{{a}_{ii}}=0\]\[\Rightarrow \]\[{{a}_{ii}}=0\]for all i Now, let A be any square matrix, then \[\frac{1}{2}(A+A')=\frac{1}{2}[A'+(A')]\] \[[\because (A+B)=A'+B']\] \[=\frac{1}{2}(A'+A)\]\[[\because (A')'=A]\] \[\Rightarrow \]\[\frac{1}{2}(A+A')\] is symmetric matrix. Also, \[\frac{1}{2}(A-A')'=\frac{1}{2}[A'-(A')']\] \[=\frac{1}{2}(A'-A)=-\frac{1}{2}(A-A')\] \[\Rightarrow \] \[\frac{1}{2}(A-A')\] is skew-symmetric matrix. Hence, option [a] is correct.
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