Let W be the vector space of 3 times 3 symmetric matrices , A in W Then , which of the following is true ? a) A^T=1 b) dimW=6 c) A^{-1}=A d) A^{-1}=A^T

beljuA 2021-01-16 Answered
Let W be the vector space of 3×3 symmetric matrices , AW Then , which of the following is true ?
a) AT=1
b) dimW=6
c) A1=A
d) A1=AT
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Expert Answer

Jayden-James Duffy
Answered 2021-01-17 Author has 91 answers
Step 1
Given that W is a vector space of 3×3 symmetric matrices.
i.e W={AM3×3|AT=A}
Let AW where A=[abcdefghi]
Then AT=A
[abcdefghi]=[adgbehcfi]
This gives b=d , g=c and h=f
So,
W={W=AM3×3|A=[abcbefcfi],a,b,c,e,f,iR}
and
[abcbefcfi]=a[100000000]+b[010100000]+c[001000100]+e[000010000]+f[000001010]+i[000000001]
Step 2
Basis of W={[100000000],[010100000],[001000100],[000010000],[000001010],[000000001]}
Therefore, dim(W)=6.
So, option (b) is correct.
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Jeffrey Jordon
Answered 2022-01-24 Author has 2047 answers

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