Gauss eliminasjon er en metode som bruker radoperasjoner på en matrise på trappeform inntil den er på redusert trappeform.
Fremgangsmåte
Start med en matrise på trappeform
Eksempel (⁕ = vilkårlig tall)
\left( \begin{array}{ccccc}
\textcolor{red}{1} & * & * & * & * \\
\textcolor{blue}{0} & \textcolor{red}{1} & * & * & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & * & *\\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & *
\end{array} \right) Steg 1: Bruk den ledende eneren i nederste rad for å få null over den
\left( \begin{array}{ccccc}
\textcolor{red}{1} & * & * & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{red}{1} & * & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & *\\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & *
\end{array} \right) Steg 2: Bruk den ledende eneren i nest nederste rad for å få null over den
\left( \begin{array}{ccccc}
\textcolor{red}{1} & * & \textcolor{blue}{0} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & *
\end{array} \right) Steg 3: Fortsett med å bruke ledende enere neste rad nedenfra for å få null over den til matrisen er på redusert trappeform
\left( \begin{array}{ccccc}
\textcolor{red}{1} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & * \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & *
\end{array} \right) + Kort video
+ Hva er en ledende ener
Dersom første tall ulik null i en rad, er en ener, er det en ledende ener.
Eksempel på ledende enere:
\left( \begin{array}{cccc}
\textcolor{red}{1} & 1 & 3 & -1 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 3 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 4 & 0
\end{array} \right),
\left( \begin{array}{ccc}
\textcolor{red}{1} & 3 & 2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1}
\end{array} \right)+ Hva er en matrise på redusert trappeform?
En matrise på redusert trappeform:
1. Første tall ulik null i hver rad, er en ener (ledende enere)
2. Rader med bare nuller samles nederst
3. Ledende enere står lengre til høyre nedover (null under ledende enere)
4. Null over ledende enere
(Matrisen er på trappeform hvis 1-3 er tilfredsstilt.)
Eksempler på matriser på redusert trappeform:
\left( \begin{array}{cccc}
\textcolor{red}{1} & \textcolor{blue}{0} & \textcolor{blue}{0} & -1 \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & 0 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 3
\end{array} \right),
\left( \begin{array}{ccc}
\textcolor{red}{1} & 3 & \textcolor{blue}{0} \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1}
\end{array} \right),
\left( \begin{array}{cc}
\textcolor{red}{1} & \textcolor{blue}{0} \\
\textcolor{blue}{0} & \textcolor{red}{1}
\end{array} \right),
\left( \begin{array}{cc}
\textcolor{red}{1} & 3\\
\textcolor{blue}{0} & \textcolor{blue}{0}
\end{array} \right)+ Eksempel 1
Bruk Gauss-Jordan eliminasjon for å få følgende matrise fra trappeform til redusert trappeform:
\left( \begin{array}{cccc}
1 & 2 & -1 & 3 \\
0 & 1 & 3 & 4 \\
0 & 0 & 1 & 2
\end{array} \right) PS: Det kan være lurt å repetere radoperasjoner før du går videre
+ Se videoløsning
Steg 1: Bruk ledende ener i nederste rad for å få null over den:
\left( \begin{array}{cccc}
\textcolor{red}{1} & 2 & -1 & 3 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 3 & 4 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 2
\end{array} \right)
\overset{R_1 + R_3 \to R_1}{\sim}
\left( \begin{array}{cccc}
\textcolor{red}{1} & 2 & \textcolor{blue}{0} & 5 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 3 & 4 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 2
\end{array} \right)
\overset{R_2 - 3 R_3 \to R_2}{\sim}
\left( \begin{array}{cccc}
\textcolor{red}{1} & 2 & \textcolor{blue}{0} & 5 \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 2
\end{array} \right) Steg 2: Bruk ledende ener i nest nederste rad for å få null over den:
\left( \begin{array}{cccc}
\textcolor{red}{1} & 2 & \textcolor{blue}{0} & 5 \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 2
\end{array} \right)
\overset{R_1 - 2 R_2 \to R_1}{\sim}
\left( \begin{array}{cccc}
\textcolor{red}{1} & \textcolor{blue}{0} & \textcolor{blue}{0} & 9 \\
\textcolor{blue}{0} & \textcolor{red}{1} & \textcolor{blue}{0} & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{red}{1} & 2
\end{array} \right) Og, vips har vi en matrise på trappeform.
+ Eksempel 2
Bruk Gauss eliminasjon for å få følgende matrise på trappeform:
\left( \begin{array}{cccc}
1 & -2 & 3 & 5 \\
0 & 1 & 4 & -2 \\
0 & 0 & 0 & 0
\end{array} \right) PS: Det kan være lurt å repetere radoperasjoner før du går videre
Steg 1: Her har vi ingen ledende ener i nederste rad som er en null-rad:
\left( \begin{array}{cccc}
\textcolor{red}{1} & -2 & 3 & 5 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 4 & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0}
\end{array} \right) Steg 2: Bruk ledende ener i nest nederste rad for å få null over den:
\left( \begin{array}{cccc}
\textcolor{red}{1} & -2 & 3 & 5 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 4 & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0}
\end{array} \right)
\overset{R_1 + 2R_2 \to R_1}{\sim}
\left( \begin{array}{cccc}
\textcolor{red}{1} & \textcolor{blue}{0} & 11 & 1 \\
\textcolor{blue}{0} & \textcolor{red}{1} & 4 & -2 \\
\textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0}
\end{array} \right)Og, vips har vi en matrise på redusert trappeform.