$$\begin{gathered} \text{ Let }A=\left(\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right).\text{ If }{u}_1\text{ and }{u}_1\text{ are column matrices such that }A{u}_1=\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right) \\ \text{ and }A{u}_2=\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)\text{ then }{u}_1+u_2\text{ is equal to } \end{gathered}$$

$$\begin{gathered} A=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right],A{u}_1=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right]\text{ and }A{u}_2=\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right] \\ \begin{array}{l}A{u}_1+A{u}_2=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}1+0\\ 0+1\\ 0+0\end{array}\right]\\ A\left(u_1+u_2\right)=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\end{array} \end{gathered}$$
$\begin{array}{l}{A}^{-1}A\left(u_1+u_2\right)=A^{-1}\left[\begin{array}{l}1\\ 1\\ 0\end{array}\right]\\ \Rightarrow u_1+u_2={\left[\begin{array}{lll}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right]}^{-1}\times \left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\end{array}$
$$\begin{gathered} \Rightarrow A^{-1}=\left[\begin{array}{r}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ -2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right] \\ \begin{array}{r}{u}_1+u_2={\left[\begin{array}{lll}1& 0& 0\\ 2& \cdot 1& 0\\ 3& 2& 1\end{array}\right]}^{-1}\times \left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\\ =\left[\begin{array}{rrr}1& 0& 0\\ -2& 1& 0\\ 1& -2& 1\end{array}\right]\times \left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\\ u_1+u_2=\left[\begin{array}{r}1+0+0\\ -2+1+0\\ 1-2+0\end{array}\right]=\left[\begin{array}{r}1\\ -1\\ -1\end{array}\right]\end{array} \end{gathered}$$