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Prove that the diagonal elements of …

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Prove that the diagonal elements of a skew symmetric matrix are all zeros
  • 1 answers

Sia ? 6 years, 2 months ago

Let {tex}A = [a_{ij}]{/tex} be a skew-symmetric matrix.
Then, {tex}a_{ij} = -a_{ij}{/tex} for all i, j
Now, put i = j, we get
{tex}\Rightarrow{/tex}{tex} a_{ii} = -a_{ii}{/tex} for all values of i
{tex}\Rightarrow{/tex}{tex} 2a_{ii} = 0{/tex}
{tex}\Rightarrow{/tex}{tex} a_{ii} = 0 {/tex} for all values of i
{tex}\therefore{/tex}  {tex}a_{11} = a_{22} = a_{33} = ... = a_{nn} = 0{/tex}
{tex}\therefore{/tex}  All the diagonal elements of a skew-symmetric matrix are zero.

Hence proved.

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