linear algebra

Question

How many different entries can a 4 * 4 skew-symmetric matrix have? An n* n skew-symmetric matrix? What if it is symmetric matrix ?

Comments

@Bikram sir

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@set2018 

Symmetric matrices  :

A square matrix A = [aij] is said to be symmetric if A′ = A, that is, [aij] = [aji] for all possible values of i and j. 

For example 

is a symmetric matrix as A′ = A

in this matrix, A[1,2] =2 and A[2,1]=2  ....  A[1,3]=3 and A[3,1] = 3  that means [aij] = [aji] for all possible values of i and j .

Skew symmetric matrices :
 A square matrix A = [aij] is said to be skew symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j.

Now, if we put i = j, we have aii = – aii. Therefore 2aii = 0 or aii = 0 for all i’s. 

This means that all the diagonal elements of a skew symmetric matrix are zero.

For example, the matrix 

is a skew symmetric matrix as B′= –B

see here  A[1,2]=e but A[2,1]= -e

Answer 1

a four by four ( 4 * 4)  skew-symmetric matrix is:

A = 

0	-a	-b	-c
a	0	-d	-e
b	d	0	-f
c	e	f	0

 Therefore 6 entries can be chosen independently

Comments

The most general form of a four by four symmetric matrix is:

A = 

a	e	f	g
e	b	h	i
f	h	c	j
g	i	h	d

Therefore 10 entries can be chosen independently.

Answer 2

In skew symmetric matrix all diagonal elements are zero and  A_ij = -A_ji which means only 2 entries must be different.

0	-2	2	-2
2	0	-2	2
-2	2	0	-2
2	-2	2	0

Comments

So if diagonal are all zero then still 12 element are there so how 2 entries can be zero?