Question

Prove that the diagonal elements of an skewsymmetric matrix are zero

Answer 1

A)

Toolbox:

A square matrix A=[aijij] is said to be skew symmetric if A'=-A that is [aij]=−[aji][aij]=−[aji] for all possible value of i and j.

To prove:

All diagonal elements of a skew-symmetric matrix are all zero.

Proof:

Let A=[aij]n×nA=[aij]n×n be a skew symmetric matrix.

⇒aij=−aji⇒aij=−aji for all i & j.

⇒aii=−aii⇒aii=−aii (Putj=i)(Putj=i)

⇒2aii=0⇒aii=0.⇒2aii=0⇒aii=0.

Thus in a skew symmetric matrix all elements along the principal diagonal are zero...

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Answer 2

in a skew symmetric matrix

aij=−aji

if elements are in diagonal then,

i=j

aii=−aii

2aii=0.

aii=0.