Let A = left[ {begin{array}{*{20}{c}}1&0&02&1 | vedclass

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(D) Given $A = \left[ {\begin{array}{*{20}{c}}1&0&0\2&1&0\3&2&1\end{array}} ight]$,$A{u_1} = \left[ {\begin{array}{*{20}{c}}1\0\0\end{array}}

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$\left[ {\begin{array}{*{20}{c}}1\{ - 1}\{ - 1}\end{array}} ight]$

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(D) Given $A = \left[ {\begin{array}{*{20}{c}}1&0&0\2&1&0\3&2&1\end{array}} ight]$,$A{u_1} = \left[ {\begin{array}{*{20}{c}}1\0\0\end{array}} ight]$,and $A{u_2} = \left[ {\begin{array}{*{20}{c}}0\1\0\end{array}} ight]$.Adding the two equations,we get:$A{u_1} + A{u_2} = \left[ {\begin{array}{*{20}{c}}1\0\0\end{array}} ight] + \left[ {\begin{array}{*{20}{c}}0\1\0\end{array}} ight]$$A(u_1 + u_2) = \left[ {\begin{array}{*{20}{c}}1\1\0\end{array}} ight]$Thus,$u_1 + u_2 = A^{-1} \left[ {\begin{array}{*{20}{c}}1\1\0\end{array}} ight]$.First,find $|A| = 1(1-0) - 0 + 0 = 1$.Next,find $adj(A)$. The cofactors are:$C_{11} = 1, C_{12} = -2, C_{13} = 1$$C_{21} = 0, C_{22} = 1, C_{23} = -2$$C_{31} = 0, C_{32} = 0, C_{33} = 1$So,$adj(A) = \left[ {\begin{array}{*{20}{c}}1&0&0\-2&1&0\1&-2&1\end{array}} ight]$.Since $|A| = 1$,$A^{-1} = adj(A)$.$u_1 + u_2 = \left[ {\begin{array}{*{20}{c}}1&0&0\-2&1&0\1&-2&1\end{array}} ight] \left[ {\begin{array}{*{20}{c}}1\1\0\end{array}} ight] = \left[ {\begin{array}{*{20}{c}}1(1)+0(1)+0(0)\-2(1)+1(1)+0(0)\1(1)-2(1)+1(0)\end{array}} ight] = \left[ {\begin{array}{*{20}{c}}1\-1\-1\end{array}} ight]$.

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