Tre typer operasjoner:
1. To rader bytter plass, Ri ↔ Rj
2. Multipliser en rad med en konstant ulik null, cRi → Ri
3. Adder en rad med multiplum av en annen rad, Ri + cRj → Ri
To matriser er radekvivalente hvis du kan få den ene ved å gjøre radoperasjoner på den andre.
+ Kort video
+ 1. To rader bytter plass
Eksempel på at rad 1 bytter plass med rad 3:
\left( \begin{array}{cccc}
\textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{2} \\
1 & 2 & 0 & 18 \\
\textcolor{blue}{3} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0}
\end{array} \right)
\overset{\textcolor{red}{R_1} \leftrightarrow \textcolor{blue}{R_3}}{\sim}
\left( \begin{array}{cccc}
\textcolor{blue}{3} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} \\
1 & 2 & 0 & 18 \\
\textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{2} \\
\end{array} \right)
Eksempel på at rad 1 bytter plass med rad 2:
\left( \begin{array}{cccc}
\textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{3} & \textcolor{red}{1} \\
\textcolor{blue}{2} & \textcolor{blue}{-4} & \textcolor{blue}{5} & \textcolor{blue}{0} \\
2 & -3 & 2 & -3
\end{array} \right)
\overset{\textcolor{red}{R_1} \leftrightarrow \textcolor{blue}{R_2}}{\sim}
\left( \begin{array}{cccc}
\textcolor{blue}{2} & \textcolor{blue}{-4} & \textcolor{blue}{5} & \textcolor{blue}{0} \\
\textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{3} & \textcolor{red}{1} \\
2 & -3 & 2 & -3
\end{array} \right)
+ 2. Multipliser rad med konstant ulik null
Eksempel på at rad 2 multipliseres med 1/2:
\left( \begin{array}{cccc}
0 & 1 & -2 & 2 \\
\textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{0} & \textcolor{red}{18} \\
3 & 0 & 0 & 0
\end{array} \right)
\overset{\textcolor{red}{R_1}/2 \to \textcolor{blue}{R_1}}{\sim}
\left( \begin{array}{cccc}
0 & 1 & -2 & 2 \\
\textcolor{blue}{\frac{1}{2}} & \textcolor{blue}{1} & \textcolor{blue}{0} & \textcolor{blue}{9} \\
3 & 0 & 0 & 0
\end{array} \right)
Eksempel på at rad 1 multipliseres med 2:
\left( \begin{array}{cccc}
\textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{3} & \textcolor{red}{1} \\
2 & -4 & 5 & 0 \\
2 & -3 & 2 & -3
\end{array} \right)
\overset{2\textcolor{red}{R_1} \to \textcolor{blue}{R_1}}{\sim}
\left( \begin{array}{cccc}
\textcolor{blue}{2} & \textcolor{blue}{-4} & \textcolor{blue}{6} & \textcolor{blue}{2} \\
2 & -4 & 5 & 0 \\
2 & -3 & 2 & -3
\end{array} \right)
+ 3. Adder en rad med multiplum av en annen rad
Eksempel på at 3 multiplisert med rad 3 legges til rad 1:
\left( \begin{array}{cccc}
\textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{2} \\
1 & 2 & 0 & 18 \\
\textcolor{blue}{3} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} \\
\end{array} \right)
\overset{\textcolor{red}{R_1} + 3\textcolor{blue}{R_3} \to \textcolor{green}{R_1}}{\sim}
\left( \begin{array}{cccc}
\textcolor{green}{9} & \textcolor{green}{1} & \textcolor{green}{-2} & \textcolor{green}{2} \\
1 & 2 & 0 & 18 \\
\textcolor{blue}{3} & \textcolor{blue}{0} & \textcolor{blue}{0} & \textcolor{blue}{0} \\
\end{array} \right)
Mellomregning:
\begin{array}{rrrrr}
\textcolor{red}{R_1}: & \textcolor{red}{0} & \textcolor{red}{1} & \textcolor{red}{-2} & \textcolor{red}{2} \\
3\textcolor{blue}{R_3}: & 9 & 0 & 0 & 0 \\ \hline
\textcolor{green}{R_1 + 3R_3}: & \textcolor{green}{9} & \textcolor{green}{1} & \textcolor{green}{-2} & \textcolor{green}{2}
\end{array}Eksempel på at -2 multiplisert med rad 1 legges til rad 2:
\left( \begin{array}{cccc}
\textcolor{blue}{1} & \textcolor{blue}{-2} & \textcolor{blue}{3} & \textcolor{blue}{1} \\
\textcolor{red}{2} & \textcolor{red}{-4} & \textcolor{red}{5} & \textcolor{red}{0} \\
2 & -3 & 2 & -3
\end{array} \right)
\overset{\textcolor{red}{R_2} - 2 \textcolor{blue}{R_1} \to \textcolor{green}{R_2}}{\sim}
\left( \begin{array}{cccc}
\textcolor{blue}{1} & \textcolor{blue}{-2} & \textcolor{blue}{3} & \textcolor{blue}{1} \\
\textcolor{green}{0} & \textcolor{green}{0} & \textcolor{green}{-1} & \textcolor{green}{-2} \\
2 & -3 & 2 & -3
\end{array} \right)
Mellomregning:
\begin{array}{rrrrr}
\textcolor{red}{R_2}: & \textcolor{red}{2} & \textcolor{red}{-4} & \textcolor{red}{5} & \textcolor{red}{0} \\
-2 \textcolor{blue}{R_1}: & -2 & 4 & -6 & -2 \\ \hline
\textcolor{green}{R_2 - 2R_1}: & \textcolor{green}{0} & \textcolor{green}{0} & \textcolor{green}{-1} & \textcolor{green}{-2}
\end{array}+ Når brukes radoperasjoner?
Radoperasjoner brukes i mange sammenhenger.
Eksempler:
– Når vil vil finne en matrise på trappeform (Gauss eliminasjon)
– Når vi vil finne en matrise på redusert trappeform (Gauss-Jordan eliminasjon)
– Når vi vil løse et ligningssett
– Når vi vil finne en invers matrise