91Ó°ÊÓ

The augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right]\) for the given matrix is represented as:

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

Perform an elementaryrow operation to produce the first augmented matrix.

Multiply row 1 by \(\frac{1}{3}\).

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

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

Perform the sum of \(1\) times row 1 and row 2 at row 2.

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

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

Write the obtained matrix \[\left[ {\begin{array}{*{20}{c}}1&{ - 3}&2&0\\0&0&0&0\end{array}} \right]\]into the equation notation.

\[\begin{array}{c}{x_1} - 3{x_2} + 2{x_3} = 0\\0 = 0\end{array}\]\(Ax = 0\)

The variables corresponding to pivot columns in the matrix are called basic variables.The other variable is called a free variable.

\({x_1}\)is a basic variable, and \({x_2}\) and \({x_3}\) are free variables.

Thus, \({x_1} = 3{x_2} - 2{x_3}\).

Sometimes the parametric form of an equation is written as\(x = s{\mathop{\rm u}\nolimits} + t{\mathop{\rm v}\nolimits} \,\,\,\left( {s,t\,{\mathop{\rm in}\nolimits} \,\mathbb{R}} \right)\).

The general solution of \(Ax = 0\) in the parametric vector form can be represented as:

\(\begin{array}{c}x = \left( {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{3{x_2} - 2{x_3}}\\{{x_2}}\\{{x_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{3{x_2}}\\{{x_2}}\\0\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{ - 2{x_3}}\\0\\{{x_3}}\end{array}} \right)\\ = {x_2}\left( {\begin{array}{*{20}{c}}3\\1\\0\end{array}} \right) + {x_3}\left( {\begin{array}{*{20}{c}}{ - 2}\\0\\1\end{array}} \right)\end{array}\)

Thus, the general solution in the parametric vector form is \(x = {x_2}\left( {\begin{array}{*{20}{c}}3\\1\\0\end{array}} \right) + {x_3}\left( {\begin{array}{*{20}{c}}{ - 2}\\0\\1\end{array}} \right)\).