If A = [aij] is a skew-symmetric matrix, then write the value of \(\d

1.	If A = [aij] is a skew-symmetric matrix, then write the value of \(\displaystyle\sum_{i}a_{ij}.\)
Answer» Given : A = [a_ij] is a skew – symmetric matrix ⇒ a_ij = - a_ji …(i) [for all values of i, j] For diagonal elements, ⇒ aii = - a_ii [for all values of i] ⇒ a_ii + a_ii= 0 ⇒ 2a_ii= 0 ⇒ a_ii= 0 …(ii) Now, \(\displaystyle\sum_{i}\)\(\displaystyle\sum_{j} a_{ij}=\) a₁₁ + a₁₂ + a₁₃...+ a₂₁ + a₂₂ + a₂₃ ... + a₃₁ + a₃₂ + a₃₃ ... = 0 + a₁₂ + a₁₃+...+ (- a₁₂) +0 + a₂₃ +... + (-a₁₃) + (-a₂₃) + 0... [from (i) and (ii)] = 0 Thus, \(\displaystyle\sum_{i}\)\(\displaystyle\sum_{j} a_{ij}=0\) Hence Proved.

If A = [aij] is a skew-symmetric matrix, then write the value of \(\displaystyle\sum_{i}a_{ij}.\)

Answer»

Given :

A = [a_ij] is a skew – symmetric matrix

⇒ a_ij = - a_ji …(i)

[for all values of i, j]

For diagonal elements,

⇒ aii = - a_ii

[for all values of i]

⇒ a_ii + a_ii= 0

⇒ 2a_ii= 0

⇒ a_ii= 0 …(ii)

Now,

\(\displaystyle\sum_{i}\)\(\displaystyle\sum_{j} a_{ij}=\) a₁₁ + a₁₂ + a₁₃...+ a₂₁ + a₂₂ + a₂₃ ... + a₃₁ + a₃₂ + a₃₃ ...

= 0 + a₁₂ + a₁₃+...+ (- a₁₂) +0 + a₂₃ +... + (-a₁₃) + (-a₂₃) + 0...

[from (i) and (ii)]

= 0

Thus,

\(\displaystyle\sum_{i}\)\(\displaystyle\sum_{j} a_{ij}=0\)

Hence Proved.

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