In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables

\(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\)

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the first row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}3&0&{ - 2}\end{array}} \right]\).

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the second row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}4&0&0\end{array}} \right]\).

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the third row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}&1\end{array}} \right]\).

So, the unknown matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}3&0&{ - 2}\\4&0&0\\1&{ - 1}&1\end{array}} \right]\).