Question

Prove that the diagonal elements of a skew symmetric matrix are all zeros.

Answer 1

Answer:

Step-by-step explanation:

Proof:

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

⇒aij=−aji for all i & j.

⇒aii=−aii (Putj=i)

⇒2aii=0⇒aii=0.

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

Answer 2

We have a square matrix , say M which is also a skew symmetric matrix

Now we have to prove that , diagonal elements of a skew symmetric matrix are always zero.

.) Let mij be the elements of the matrix M , where i is donating rows and j is donating column.

Now , matrix M is a skew symmetric matrix

⇒ mij = −mji for all i & j.

⇒ mij + mji = 0

.)  Now condition for diagonal elements is i = j , hence mii or mjj represents the diagonal elements

=> mii + mii = 0

=> 2*mii = 0

=> mii = 0 = mjj

Hence every diagonal element is zero  

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