Find the general solutions of the systems whose augmented matrices are given

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

A basic principle states that row operations do not affect the solution set of alinear system. Perform an elementary row operation to produce the first augmented matrix.

Multiply row \(1\) by \(\frac{1}{3}\) to get 1 in the first equation.

\(\left[ {\begin{array}{*{20}{c}}1&{ - \frac{4}{3}}&{\frac{2}{3}}&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\)

Perform an elementary row operation to produce a second augmented matrix.

Replace row 2 by adding\(9\)times row 1 to row \(2\).

\(\left[ {\begin{array}{*{20}{c}}1&{ - \frac{4}{3}}&{\frac{2}{3}}&0\\0&0&0&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\)

Perform an elementary row operationto produce a third augmented matrix.

Replace row \(3\) by adding \(6\)times row \(1\) to row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{ - \frac{4}{3}}&{\frac{2}{3}}&0\\0&0&0&0\\0&0&0&0\end{array}} \right]\)

To obtain the general solution of the system, you have to convert the augmented matrix into the system of equations.

Write the obtained matrix into the equation notation.

\({x_1} - \frac{4}{3}{x_2} + \frac{2}{3}{x_3} = 0\)

Thus, the general solution of the system is

\({x_1} = \frac{4}{3}{x_2} - \frac{2}{3}{x_3}\)

\({x_2}\)is free.

\({x_3}\) is free.