Question

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

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

Short answer

\(\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\{ - 2}&1\\1&0\end{array}} \right]\)

Step-by-step solution

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the first row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}\end{array}} \right]\).

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the second row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}{ - 2}&1\end{array}} \right]\).

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the third row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}1&0\end{array}} \right]\).

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