1 An Example for Gaussian Elimination

Consider the real matrix

A(t) := (\begin{matrix}1 & 1 & 1 \\ t & 2t & 2 \\ t + 1 & 0 & 2t \end{matrix}) ,

which depends on a real valued parameter t. We want to find all solutions of the homogenous linear system defined by A depending on the parameter t.

Therefore, we use the Algorithm Gauss:

A(t) = (\begin{matrix}1 & 1 & 1 \\ t & 2t & 2 \\ t + 1 & 0 & 2t \end{matrix}) ⟵^{−t}^{+} ⟵^{−(t+1)}^{+}  ⇝ (\begin{matrix}1 & 1 & 1 \\ 0 & t & 2 − t \\ 0 & − t − 1 & t − 1 \end{matrix}) _{⟵^{+} | · (−1)}

⇝ (\begin{matrix}1 & 1 & 1 \\ 0 & t & 2 − t \\ 0 & 1 & − 1 \end{matrix}) ⟵ ⟵ ⟵^{−t}^{+} | : 2

⇝ (\begin{matrix}1 & 1 & 1 \\ 0 & 1 & −1 \\ 0 & 0 & 1 \end{matrix}) .

Conclusion

We have learned that the matrix A defined above is regular for all real valued t, and we hopefully also have learned how to use the gauss package.