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Reducerad radtrappstegsform för $$$\left[\begin{array}{ccc}3 & -4 & 2\\1 & 6 & 8\\2 & 7 & 9\end{array}\right]$$$
Relaterade kalkylatorer: Kalkylator för Gauss-Jordan-eliminering, Inversmatriskalkylator
Din inmatning
Bestäm den reducerade radtrappstegsformen för $$$\left[\begin{array}{ccc}3 & -4 & 2\\1 & 6 & 8\\2 & 7 & 9\end{array}\right]$$$.
Lösning
Dividera rad $$$1$$$ med $$$3$$$: $$$R_{1} = \frac{R_{1}}{3}$$$.
$$$\left[\begin{array}{ccc}1 & - \frac{4}{3} & \frac{2}{3}\\1 & 6 & 8\\2 & 7 & 9\end{array}\right]$$$
Subtrahera rad $$$1$$$ från rad $$$2$$$: $$$R_{2} = R_{2} - R_{1}$$$.
$$$\left[\begin{array}{ccc}1 & - \frac{4}{3} & \frac{2}{3}\\0 & \frac{22}{3} & \frac{22}{3}\\2 & 7 & 9\end{array}\right]$$$
Subtrahera rad $$$1$$$ multiplicerad med $$$2$$$ från rad $$$3$$$: $$$R_{3} = R_{3} - 2 R_{1}$$$.
$$$\left[\begin{array}{ccc}1 & - \frac{4}{3} & \frac{2}{3}\\0 & \frac{22}{3} & \frac{22}{3}\\0 & \frac{29}{3} & \frac{23}{3}\end{array}\right]$$$
Multiplicera rad $$$2$$$ med $$$\frac{3}{22}$$$: $$$R_{2} = \frac{3 R_{2}}{22}$$$.
$$$\left[\begin{array}{ccc}1 & - \frac{4}{3} & \frac{2}{3}\\0 & 1 & 1\\0 & \frac{29}{3} & \frac{23}{3}\end{array}\right]$$$
Lägg till rad $$$2$$$ multiplicerad med $$$\frac{4}{3}$$$ till rad $$$1$$$: $$$R_{1} = R_{1} + \frac{4 R_{2}}{3}$$$.
$$$\left[\begin{array}{ccc}1 & 0 & 2\\0 & 1 & 1\\0 & \frac{29}{3} & \frac{23}{3}\end{array}\right]$$$
Subtrahera rad $$$2$$$ multiplicerad med $$$\frac{29}{3}$$$ från rad $$$3$$$: $$$R_{3} = R_{3} - \frac{29 R_{2}}{3}$$$.
$$$\left[\begin{array}{ccc}1 & 0 & 2\\0 & 1 & 1\\0 & 0 & -2\end{array}\right]$$$
Dividera rad $$$3$$$ med $$$-2$$$: $$$R_{3} = - \frac{R_{3}}{2}$$$.
$$$\left[\begin{array}{ccc}1 & 0 & 2\\0 & 1 & 1\\0 & 0 & 1\end{array}\right]$$$
Subtrahera rad $$$3$$$ multiplicerad med $$$2$$$ från rad $$$1$$$: $$$R_{1} = R_{1} - 2 R_{3}$$$.
$$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 1\\0 & 0 & 1\end{array}\right]$$$
Subtrahera rad $$$3$$$ från rad $$$2$$$: $$$R_{2} = R_{2} - R_{3}$$$.
$$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$
Svar
Den reducerade radtrappstegsformen är $$$\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$A.