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$ $
Opg 1: Elementære udregninger Der er givet fire matricer
$$\mA=\begin{matr}{rrr} -1&0&5\\\\ 2&-1&5\\\\ 0&2&-1 \end{matr}\,,\,\,
\mB=\begin{matr}{rr} 2&1\\\\0&0\\\\ 1&-2\end{matr}\,,\,\,
\mC=\begin{matr}{rrr} 4&0&3\\\\ 1&-2&3\end{matr}\,\,\,\mathrm{og}\,\,\,
\mD=\begin{matr}{rrr} 1&1&0\\\\ 1&-1&0\\\\ 2&1&-2 \end{matr}\,.$$
A
Udregn $\,4\mathbf C\,$ og $-\mathbf C\,.$
B
Udregn $\,\mathbf A+\mathbf D\,$ og $\,3\mathbf A-2\mathbf D\,.$
C
Udregn $\,-2\mathbf B+3\mathbf B\,\,.$
###Opg 2: Matrixprodukter
Der er givet matricerne
$$\mathbf A=\begin{matr}{rr} 2&1\\\\0&3\\\\ 1&-2\end{matr}\,,\,\,
\mathbf B=\begin{matr}{rrr} 4&0&3\\\\ 1&-2&3\end{matr}\,\,\,\mathrm{og}\,\,\,\mC=\begin{matr}{rrr} 1&1&0\\\\ 1&-1&0\\\\ 2&1&-2 \end{matr}\,$$
samt de to vektorer
$$\mathbf u= \begin{matr}{r} 1\\\\-2\\\\-1\end{matr}\,\,\,\mathrm{og}\,\,\,\mathbf v= \begin{matr}{r} 1\\\\2\end{matr}\,.$$
A
Udregn matrix-vektorprodukterne $\,\mA\mv\,$ og $\,\mB\mathbf u\,.$
Show answer
$\begin{matr}{r} 4\\6\\-3\end{matr}\,$ og $\begin{matr}{r} 1\\2\end{matr}\,$
B
Hvilken form forventer du at matrix-matrixprodukterne $\,\mA\mB\,$ og $\,\mB\mathbf A\,$ har? Udregn dem!
Show answer
$\begin{matr}{rrr} 9&-2&9\\3&-6&9\\2&4&-3\end{matr}\,$ og $\begin{matr}{rr} 11&-2\\5&-11\end{matr}\,.$
Opg 3: Kvadratiske matricer Der er givet to matricer $\,\mathbf A=\begin{matr}{rr} 1&0\\3&2\end{matr}\,$ og $\,\mathbf B=\begin{matr}{rr} -1&4\\1&-2\end{matr}\,$ samt enhedsmatricen $\,\mathbf E=\begin{matr}{rr} 1&0\\0&1\end{matr}\,.$
A
Find produkterne $\,\mA\mathbf E\,$ og $\,\mB\mathbf E\,.$
Show hint
Du behøver ikke lave udregninger. Hvorfor?
B
Udregn produkterne $\,\mA\mathbf B\,$ og $\,\mB\mathbf A\,$ og kommentér!
Show hint
Opfylder matrix-matrix produkt den kommutative lov ?
Opg 4: Determinanter Givet matricerne
$$\mathbf A=\begin{matr}{rr} -3&-2\\\\5&4\end{matr}\,,\,\,\mathbf B=\begin{matr}{rrr} 1&2&3\\\\4&5&6\\\\7&8&9\end{matr}\,\,\,\mathrm{og}\,\,\,\mC=\begin{matr}{rrrr} 1 & 0 & 1 & 1 \\\\ 0 & 2 & 2 & 4 \\\\ 1 & 1 & 0 & 0 \\\\ 1 & 1 & 2 & 0 \end{matr}\,.$$
A
Udregn det$(\mA)\,$ og det$(\mB)\,$ ved håndregning, og det$(\mC)\,$ med dit matematikværktøj.
Show answer
$-2\,,\,\,0\,$ og $\,12\,.$
B
Prøv med dit matematikværktøj at finde den inverse matrix til hver af de tre matricer.
Opg 5: Inverse matricer Givet matricerne
$$
\mA = \begin{matr}{rr} 2 & 3 \\\\ 1 & 1 \end{matr} \, , \quad \mB = \begin{matr}{rr} 1 & 0 \\\\ 4 & 1 \end{matr} \, , \quad \mC = \begin{matr}{rr} -1 & 3 \\\\ 1 & -2 \end{matr} \quad \mathrm{og} \quad \mD = \begin{matr}{rr} 1 & 0 \\\\ -4 & 1 \end{matr}\,.$$
A
Find determinanterne af $\,\mA\,$ og $\,\mB\,$ og gør rede for at de begge har en invers matrix.
B
Udregn $\,\mA\mC\,$ og $\,\mB\mD \,$ og kommentér!
Show answer
\begin{equation}
\mA\mC = \begin{matr}{rr} 1 & 0 \\ 0 & 1 \end{matr},\,\,
\mB\mD = \begin{matr}{rr} 1 & 0 \\ 0 & 1 \end{matr}\,.
\end{equation}
C
Find $ \mA^{-1} $ og $ \mB^{-1} \,$ uden at lave nye beregninger!
Show answer
$\mA^{-1} = \mC$ , idet $\mA\mC=\mE$ og $\mB^{-1} = \mD$ , idet $\mB\mD=\mE$ .
Opg 6: En matrixligning løst med invers matrix Info: Hvis en $\,2\times 2\,$ matrix $\,\mA = \begin{matr}{rr} a & b \\ c & d \end{matr} \,$ har en invers matrix, kan den inverse matrix findes ved formlen:
$$
\mA^{-1} = \frac 1{\mathrm{det}(\mA)}\,\begin{matr}{rr} d & -b \\\\ -c & a \end{matr}\,.$$
Der er givet matricerne
$$ \mA = \begin{matr}{rr} 2 & 1 \\\\ 1 & 1 \end{matr}\,\,\,\mathrm{og}\,\,\,\mB = \begin{matr}{rr} 1 & 0 \\\\ 2 & -2 \end{matr} \,.$$
A
Find $ \mA^{-1} $ og løs matrixligningen $\,\mA\mathbf X=\mB\,.$
Show answer
$\mA^{-1} = \begin{matr}{rr} 1 & -1 \\ -1 & 2 \end{matr}\,$ og $\,\mathbf X =\begin{matr}{rr} -1 & 2 \\ 3 & -4 \end{matr}\,.$
B
Tjek resultaterne med dit matematikværktøj.
