Let Omega be the set of all 3 times 3 symmetr | vedclass

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(A, D, C) $3 	imes 3$ symmetric matrix $A$ is determined by its $6$ entries: $a_{11}, a_{22}, a_{33}, a_{12}, a_{13}, a_{23}$.$1.$ Total entries are

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$(A, B, B)$

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(A, D, C) $3 	imes 3$ symmetric matrix $A$ is determined by its $6$ entries: $a_{11}, a_{22}, a_{33}, a_{12}, a_{13}, a_{23}$.$1.$ Total entries are $9$. Given $5$ ones and $4$ zeros. Since $A$ is symmetric,$a_{12}=a_{21}, a_{13}=a_{31}, a_{23}=a_{32}$.Let $k$ be the number of $1$s on the diagonal. The number of $1$s off-diagonal must be $(5-k)$. Since off-diagonal entries come in pairs,$(5-k)$ must be even. Thus $k$ must be odd ($1$ or $3$).If $k=3$,all diagonal entries are $1$. We need $5-3=2$ ones off-diagonal. We choose $1$ pair out of $3$ pairs to be $1$ (i.e.,$3$ matrices).If $k=1$,one diagonal entry is $1$. We need $5-1=4$ ones off-diagonal. We choose $2$ pairs out of $3$ pairs to be $1$ (i.e.,$3 	imes 3 = 9$ matrices).Total matrices $= 3 + 9 = 12$.$2.$ $A$ unique solution exists if $|A| 
eq 0$. Evaluating the $12$ matrices,we find $6$ matrices have $|A| 
eq 0$.$3.$ For the remaining $6$ matrices where $|A| = 0$,we check for inconsistency. By testing the augmented matrix $[A|B]$,we find $2$ matrices are inconsistent.

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