# skew-symmetric metrix

• Oct 23rd 2010, 02:34 AM
1234567
skew-symmetric metrix
Hi i need some help solving the following problem:

Attachment 19424

What i know:

The following does not hold for p=2 since an inverse does not hold.

but find it diffecult to show that this holds for primes p> 3.

Thanks
• Oct 23rd 2010, 04:51 AM
tonio
Quote:

Originally Posted by 1234567
Hi i need some help solving the following problem:

Attachment 19424

What i know:

The following does not hold for p=2 since an inverse does not hold.

but find it diffecult to show that this holds for primes p> 3.

Thanks

For a prime \$\displaystyle p\geq 3\$ we have that \$\displaystyle 1\neq -1\$ and thus it's easy to show that \$\displaystyle dim S_n+dim A_n=dim M_n\,,\,\,M_n=\$ all the square nxn matrices over the field...complete the proof now.

Tonio
• Oct 24th 2010, 03:57 AM
1234567
how would one use \$\displaystyle dim S_n+dim A_n=dim M_n\,,\,\,M_n=\$ all the square nxn matrices over the field,
I thought you hav to show

1. \$\displaystyle S_n+A_n= M_n\,,\$
2. \$\displaystyle S_n n A_n= {0}\,,\$
• Oct 24th 2010, 04:18 AM
tonio
Quote:

Originally Posted by 1234567
how would one use \$\displaystyle dim S_n+dim A_n=dim M_n\,,\,\,M_n=\$ all the square nxn matrices over the field,
I thought you hav to show

1. \$\displaystyle S_n+A_n= M_n\,,\$
2. \$\displaystyle S_n n A_n= {0}\,,\$

The second condition is trivial, so \$\displaystyle \dim S_n+\dim A_n=\dim(S_n+A_n)-\dim(S_n\cap A_n)=\dim(S_n+A_n)\$ , and thus

proving what I told you we get \$\displaystyle S_n+A_n=M_n\$

Tonio