&\begin{aligned} & \begin{array}{l} A=\left[\begin{array}{lll} 1 & 2 & 2 \\\\ 3 & 2 & 3 \\\\ 1 & 1 & 2 \end{array}\right] \text { and } A^{-1}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \end{array}\right] \\\\ |A|=1(4-3)-2(6-3)+2(3-2)=1-6+2=-3\begin{array}{l} \end{array} \\\\ C_{11}=\left|\begin{array}{ll} 2 & 3 \\\\ 1 & 2 \end{array}\right|=1 \\\\ C_{12}=-\left|\begin{array}{ll} 3 & 3 \\\\ 1 & 2 \end{array}\right|=-3 \\\\ C_{13}=\left|\begin{array}{ll} 3 & 2 \\\\ 1 & 1 \end{array}\right|=1 \\\\ C_{21}=-\left|\begin{array}{ll} 2 & 2 \\\\ 1 & 2 \end{array}\right|=-2 \\\\ C_{22}=\left|\begin{array}{ll} 1 & 2 \\\\ 1 & 2 \end{array}\right|=0 \\\\ C_{23}=-\left|\begin{array}{ll} 1 & 2 \\\\ 1 & 1 \end{array}\right|=1 \end{array} \end{aligned} \end{aligned} $$

$$ \begin{aligned} & C_{31}=\left|\begin{array}{ll} 2 & 2 \\\\ 2 & 3 \end{array}\right|=2, C_{32}=-\left|\begin{array}{ll} 1 & 2 \\\\ 3 & 3 \end{array}\right|=3 \\\\ & C_{33}=\left|\begin{array}{ll} 1 & 2 \\\\ 3 & 2 \end{array}\right|=-4 \\\\ & \operatorname{adj} A=\left[\begin{array}{ccc} 1 & -3 & 1 \\\\ -2 & 0 & 1 \\\\ 2 & 3 & -4 \end{array}\right]^T=\left[\begin{array}{ccc} 1 & -2 & 2 \\\\ -3 & 0 & 3 \\\\ 1 & 1 & -4 \end{array}\right] \\\\ & \therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{-3}\left[\begin{array}{ccc} 1 & -2 & 2 \\\\ -3 & 0 & 3 \\\\ 1 & 1 & -4 \end{array}\right] \end{aligned} $$

Now,

$$ \sum_{\substack{1 \leq i \leq 3 \\\\ 1 \leq j \leq 3}} a_{i j}=\left(-\frac{1}{3}\right) $$

$$ \begin{aligned} & {[1+(-2)+2+(-3)+0+3+1+1+(-4)]} \\\\ & \quad=\left(-\frac{1}{3}\right)(-1)=\frac{1}{3} \end{aligned} $$

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