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

beljuA

Answered question

2021-01-16

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

Answer & Explanation

Jayden-James Duffy

Jayden-James Duffy

Skilled2021-01-17Added 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.
Jeffrey Jordon

Jeffrey Jordon

Expert2022-01-24Added 2605 answers

Answer is given below (on video)

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