# 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

Let W be the vector space of $3×3$ symmetric matrices , $A\in W$ Then , which of the following is true ?
a) ${A}^{T}=1$
b) $dimW=6$
c) ${A}^{-1}=A$
d) ${A}^{-1}={A}^{T}$
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Jayden-James Duffy
Step 1
Given that W is a vector space of $3×3$ symmetric matrices.
i.e $W=\left\{A\in {M}_{3×3}|{A}^{T}=A\right\}$
Let $A\in W$ where $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$
Then ${A}^{T}=A$
$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=\left[\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right]$
This gives b=d , g=c and h=f
So,
$W=\left\{W=A\in {M}_{3×3}|A=\left[\begin{array}{ccc}a& b& c\\ b& e& f\\ c& f& i\end{array}\right],a,b,c,e,f,i\in R\right\}$
and
$\left[\begin{array}{ccc}a& b& c\\ b& e& f\\ c& f& i\end{array}\right]=a\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+b\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]+c\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]+e\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]+f\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]+i\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$
Step 2
Basis of $W=\left\{\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\right\}$
Therefore, dim(W)=6.
So, option (b) is correct.
Jeffrey Jordon