====== Brückenkurs "Lineare Algebra" ======
Termine jeweils 12:20 bis 13:45 im H21:
* Freitag 13. November 2020, online auf Teams
* Freitag 20. November 2020
* Freitag 27. November 2020
===== Unterlagen =====
Die Theorie und Aufgaben erhalten Sie auf Papier, die Lösungen nur online.
* **Achtung: Nicht ausdrucken**, ist in Entwicklung, ändert sich laufend... {{ :lehrkraefte:blc:linalg20:linearealgebra-sv.pdf |Theorie, Aufgaben und Lösungen}}.
* {{ :lehrkraefte:blc:linalg20:lineare-abbildung-grid.ggb |GeoGebra-Datei zur Veranschaulichung einer lineare Abbildung der Ebene}}
* {{ :lehrkraefte:blc:linalg20:linearealgebra-l1.pdf |Notizen der ersten Lektion vom 13.11.2020}}
===== Übungsaufgaben =====
miniaufgabe.js
Hinweis: Die Aufgaben sind automatisch generiert und ändern sich beim Neu-Laden der Seite.
==== Matrix-Vektor Multiplikation ====
Berechnen Sie:miniAufgabe("#exomatrixvectormultiplikation","#solmatrixvectormultiplikation",
[["$\\begin{pmatrix}0 & -3\\\\\n-5 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0\\\\\n-6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}0 & -3\\\\\n-5 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0\\\\\n-6\\\\\n\\end{pmatrix} = \\begin{pmatrix}18\\\\\n-24\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}3 & 0\\\\\n-3 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5\\\\\n2\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}3 & 0\\\\\n-3 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5\\\\\n2\\\\\n\\end{pmatrix} = \\begin{pmatrix}15\\\\\n-13\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}0 & -6\\\\\n0 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n-1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}0 & -6\\\\\n0 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n-1\\\\\n\\end{pmatrix} = \\begin{pmatrix}6\\\\\n-3\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}1 & 3\\\\\n-3 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1\\\\\n-4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}1 & 3\\\\\n-3 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1\\\\\n-4\\\\\n\\end{pmatrix} = \\begin{pmatrix}-11\\\\\n-15\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}6 & -6\\\\\n2 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-4\\\\\n-1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}6 & -6\\\\\n2 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-4\\\\\n-1\\\\\n\\end{pmatrix} = \\begin{pmatrix}-18\\\\\n-10\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}5 & 4\\\\\n4 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-3\\\\\n-5\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}5 & 4\\\\\n4 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-3\\\\\n-5\\\\\n\\end{pmatrix} = \\begin{pmatrix}-35\\\\\n-17\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-3 & -6\\\\\n3 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1\\\\\n6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-3 & -6\\\\\n3 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1\\\\\n6\\\\\n\\end{pmatrix} = \\begin{pmatrix}-33\\\\\n-3\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}6 & 3\\\\\n3 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6\\\\\n1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}6 & 3\\\\\n3 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6\\\\\n1\\\\\n\\end{pmatrix} = \\begin{pmatrix}39\\\\\n22\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-4 & 6\\\\\n4 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-4 & 6\\\\\n4 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n6\\\\\n\\end{pmatrix} = \\begin{pmatrix}60\\\\\n-54\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}5 & -6\\\\\n-5 & 6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5\\\\\n0\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}5 & -6\\\\\n-5 & 6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5\\\\\n0\\\\\n\\end{pmatrix} = \\begin{pmatrix}25\\\\\n-25\\\\\n\\end{pmatrix}$"]],
"
");
Berechnen Sie:miniAufgabe("#exomatrixvectormultiplikation2D","#solmatrixvectormultiplikation2D",
[["$\\begin{pmatrix}1 & 1 & 6\\\\\n-1 & 0 & -4\\\\\n-6 & 3 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6\\\\\n-2\\\\\n4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}1 & 1 & 6\\\\\n-1 & 0 & -4\\\\\n-6 & 3 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6\\\\\n-2\\\\\n4\\\\\n\\end{pmatrix} = \\begin{pmatrix}28\\\\\n-22\\\\\n-46\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-4 & -3 & 2\\\\\n3 & 4 & 3\\\\\n-1 & -5 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n-1\\\\\n3\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-4 & -3 & 2\\\\\n3 & 4 & 3\\\\\n-1 & -5 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n-1\\\\\n3\\\\\n\\end{pmatrix} = \\begin{pmatrix}-3\\\\\n14\\\\\n2\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-2 & 0 & -1\\\\\n3 & -1 & 2\\\\\n5 & 1 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5\\\\\n0\\\\\n1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-2 & 0 & -1\\\\\n3 & -1 & 2\\\\\n5 & 1 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5\\\\\n0\\\\\n1\\\\\n\\end{pmatrix} = \\begin{pmatrix}9\\\\\n-13\\\\\n-31\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-3 & -1 & 1\\\\\n2 & 5 & -6\\\\\n3 & -2 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0\\\\\n2\\\\\n2\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-3 & -1 & 1\\\\\n2 & 5 & -6\\\\\n3 & -2 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0\\\\\n2\\\\\n2\\\\\n\\end{pmatrix} = \\begin{pmatrix}0\\\\\n-2\\\\\n-4\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}0 & 3 & -1\\\\\n-1 & -1 & 4\\\\\n2 & -1 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n5\\\\\n-4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}0 & 3 & -1\\\\\n-1 & -1 & 4\\\\\n2 & -1 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n5\\\\\n-4\\\\\n\\end{pmatrix} = \\begin{pmatrix}19\\\\\n-24\\\\\n-3\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}5 & 3 & 5\\\\\n-2 & -3 & 1\\\\\n4 & -1 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5\\\\\n-1\\\\\n-6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}5 & 3 & 5\\\\\n-2 & -3 & 1\\\\\n4 & -1 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5\\\\\n-1\\\\\n-6\\\\\n\\end{pmatrix} = \\begin{pmatrix}-58\\\\\n7\\\\\n-31\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}6 & 5 & 1\\\\\n-2 & -3 & 3\\\\\n-4 & -2 & 5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n-1\\\\\n6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}6 & 5 & 1\\\\\n-2 & -3 & 3\\\\\n-4 & -2 & 5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n-1\\\\\n6\\\\\n\\end{pmatrix} = \\begin{pmatrix}-35\\\\\n33\\\\\n56\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-6 & -1 & 0\\\\\n-6 & -4 & -6\\\\\n1 & 1 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n3\\\\\n2\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-6 & -1 & 0\\\\\n-6 & -4 & -6\\\\\n1 & 1 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6\\\\\n3\\\\\n2\\\\\n\\end{pmatrix} = \\begin{pmatrix}33\\\\\n12\\\\\n-1\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}4 & 3 & -3\\\\\n5 & -6 & 1\\\\\n-3 & 4 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-2\\\\\n-3\\\\\n-6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}4 & 3 & -3\\\\\n5 & -6 & 1\\\\\n-3 & 4 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-2\\\\\n-3\\\\\n-6\\\\\n\\end{pmatrix} = \\begin{pmatrix}1\\\\\n2\\\\\n12\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-3 & 2 & -3\\\\\n2 & 5 & -6\\\\\n0 & 0 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n-3\\\\\n-1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-3 & 2 & -3\\\\\n2 & 5 & -6\\\\\n0 & 0 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3\\\\\n-3\\\\\n-1\\\\\n\\end{pmatrix} = \\begin{pmatrix}-12\\\\\n-3\\\\\n-3\\\\\n\\end{pmatrix}$"]],
"
");
==== Matrix-Matrix Multiplikation ====
Berechnen Sie:miniAufgabe("#exomatrixmatrixmultiplikation3D","#solmatrixmatrixmultiplikation3D",
[["$\\begin{pmatrix}-2 & -4\\\\\n-6 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & -5\\\\\n1 & 3\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-2 & -4\\\\\n-6 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & -5\\\\\n1 & 3\\\\\n\\end{pmatrix} = \\begin{pmatrix}-8 & -2\\\\\n-12 & 30\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}2 & -4\\\\\n0 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1 & 1\\\\\n-1 & -6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}2 & -4\\\\\n0 & 2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1 & 1\\\\\n-1 & -6\\\\\n\\end{pmatrix} = \\begin{pmatrix}2 & 26\\\\\n-2 & -12\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}6 & 3\\\\\n4 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & 4\\\\\n-2 & -4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}6 & 3\\\\\n4 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & 4\\\\\n-2 & -4\\\\\n\\end{pmatrix} = \\begin{pmatrix}6 & 12\\\\\n12 & 24\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}3 & 3\\\\\n5 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1 & -6\\\\\n3 & -4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}3 & 3\\\\\n5 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-1 & -6\\\\\n3 & -4\\\\\n\\end{pmatrix} = \\begin{pmatrix}6 & -30\\\\\n-8 & -26\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-1 & -1\\\\\n4 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & -4\\\\\n-3 & 1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-1 & -1\\\\\n4 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & -4\\\\\n-3 & 1\\\\\n\\end{pmatrix} = \\begin{pmatrix}-3 & 3\\\\\n30 & -18\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}0 & -1\\\\\n5 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6 & 2\\\\\n-2 & 4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}0 & -1\\\\\n5 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6 & 2\\\\\n-2 & 4\\\\\n\\end{pmatrix} = \\begin{pmatrix}2 & -4\\\\\n-20 & -10\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}2 & 0\\\\\n2 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}4 & 3\\\\\n4 & 6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}2 & 0\\\\\n2 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}4 & 3\\\\\n4 & 6\\\\\n\\end{pmatrix} = \\begin{pmatrix}8 & 6\\\\\n-16 & -30\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-1 & 5\\\\\n0 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & -4\\\\\n4 & 0\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-1 & 5\\\\\n0 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & -4\\\\\n4 & 0\\\\\n\\end{pmatrix} = \\begin{pmatrix}14 & 4\\\\\n-12 & 0\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-5 & 6\\\\\n-4 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & 5\\\\\n1 & -5\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-5 & 6\\\\\n-4 & -6\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}6 & 5\\\\\n1 & -5\\\\\n\\end{pmatrix} = \\begin{pmatrix}-24 & -55\\\\\n-30 & 10\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}4 & -6\\\\\n1 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6 & 3\\\\\n0 & 4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}4 & -6\\\\\n1 & -2\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-6 & 3\\\\\n0 & 4\\\\\n\\end{pmatrix} = \\begin{pmatrix}-24 & -12\\\\\n-6 & -5\\\\\n\\end{pmatrix}$"]],
"
");
Berechnen Sie:miniAufgabe("#exomatrixmatrixmultiplikation4D","#solmatrixmatrixmultiplikation4D",
[["$\\begin{pmatrix}2 & 3 & -4\\\\\n2 & 4 & 0\\\\\n0 & 0 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5 & -4 & -5\\\\\n3 & -5 & 3\\\\\n-6 & 1 & -6\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}2 & 3 & -4\\\\\n2 & 4 & 0\\\\\n0 & 0 & -1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-5 & -4 & -5\\\\\n3 & -5 & 3\\\\\n-6 & 1 & -6\\\\\n\\end{pmatrix} = \\begin{pmatrix}23 & -27 & 23\\\\\n2 & -28 & 2\\\\\n6 & -1 & 6\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-4 & 4 & -4\\\\\n-6 & 4 & 2\\\\\n2 & -1 & -4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0 & 6 & -2\\\\\n-3 & -1 & 2\\\\\n2 & 4 & -5\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-4 & 4 & -4\\\\\n-6 & 4 & 2\\\\\n2 & -1 & -4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0 & 6 & -2\\\\\n-3 & -1 & 2\\\\\n2 & 4 & -5\\\\\n\\end{pmatrix} = \\begin{pmatrix}-20 & -44 & 36\\\\\n-8 & -32 & 10\\\\\n-5 & -3 & 14\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}3 & -6 & 6\\\\\n-6 & -1 & 0\\\\\n-5 & 6 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1 & 4 & -4\\\\\n-4 & 6 & -4\\\\\n-2 & 6 & -1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}3 & -6 & 6\\\\\n-6 & -1 & 0\\\\\n-5 & 6 & 0\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1 & 4 & -4\\\\\n-4 & 6 & -4\\\\\n-2 & 6 & -1\\\\\n\\end{pmatrix} = \\begin{pmatrix}15 & 12 & 6\\\\\n-2 & -30 & 28\\\\\n-29 & 16 & -4\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}2 & -1 & -4\\\\\n-6 & 0 & 0\\\\\n-3 & 1 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5 & -5 & 4\\\\\n3 & -3 & 4\\\\\n2 & -4 & -2\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}2 & -1 & -4\\\\\n-6 & 0 & 0\\\\\n-3 & 1 & 3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}5 & -5 & 4\\\\\n3 & -3 & 4\\\\\n2 & -4 & -2\\\\\n\\end{pmatrix} = \\begin{pmatrix}-1 & 9 & 12\\\\\n-30 & 30 & -24\\\\\n-6 & 0 & -14\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}4 & 0 & 2\\\\\n2 & -4 & -1\\\\\n-1 & -2 & 5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3 & 5 & 2\\\\\n5 & 3 & -3\\\\\n4 & 5 & 5\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}4 & 0 & 2\\\\\n2 & -4 & -1\\\\\n-1 & -2 & 5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3 & 5 & 2\\\\\n5 & 3 & -3\\\\\n4 & 5 & 5\\\\\n\\end{pmatrix} = \\begin{pmatrix}20 & 30 & 18\\\\\n-18 & -7 & 11\\\\\n7 & 14 & 29\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-4 & -3 & 5\\\\\n-4 & 5 & -2\\\\\n6 & 0 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1 & -1 & 3\\\\\n-4 & -3 & 4\\\\\n1 & -5 & 3\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-4 & -3 & 5\\\\\n-4 & 5 & -2\\\\\n6 & 0 & -5\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}1 & -1 & 3\\\\\n-4 & -3 & 4\\\\\n1 & -5 & 3\\\\\n\\end{pmatrix} = \\begin{pmatrix}13 & -12 & -9\\\\\n-26 & -1 & 2\\\\\n1 & 19 & 3\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-3 & 5 & -4\\\\\n0 & -3 & 4\\\\\n0 & -1 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0 & 2 & 3\\\\\n-2 & -4 & -2\\\\\n5 & 6 & -4\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-3 & 5 & -4\\\\\n0 & -3 & 4\\\\\n0 & -1 & -3\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}0 & 2 & 3\\\\\n-2 & -4 & -2\\\\\n5 & 6 & -4\\\\\n\\end{pmatrix} = \\begin{pmatrix}-30 & -50 & -3\\\\\n26 & 36 & -10\\\\\n-13 & -14 & 14\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-2 & 3 & 6\\\\\n4 & 2 & -6\\\\\n-1 & 2 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3 & 6 & -6\\\\\n-4 & -6 & -2\\\\\n1 & -4 & -5\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-2 & 3 & 6\\\\\n4 & 2 & -6\\\\\n-1 & 2 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}3 & 6 & -6\\\\\n-4 & -6 & -2\\\\\n1 & -4 & -5\\\\\n\\end{pmatrix} = \\begin{pmatrix}-12 & -54 & -24\\\\\n-2 & 36 & 2\\\\\n-10 & -22 & -3\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}-6 & 5 & -1\\\\\n-1 & -4 & 4\\\\\n2 & -5 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-2 & 1 & -3\\\\\n2 & 6 & 3\\\\\n5 & -3 & 1\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}-6 & 5 & -1\\\\\n-1 & -4 & 4\\\\\n2 & -5 & 4\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}-2 & 1 & -3\\\\\n2 & 6 & 3\\\\\n5 & -3 & 1\\\\\n\\end{pmatrix} = \\begin{pmatrix}17 & 27 & 32\\\\\n14 & -37 & -5\\\\\n6 & -40 & -17\\\\\n\\end{pmatrix}$"], ["$\\begin{pmatrix}4 & 5 & -4\\\\\n-3 & -3 & 2\\\\\n-5 & -3 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & 3 & -5\\\\\n-6 & -1 & -4\\\\\n6 & 5 & 3\\\\\n\\end{pmatrix}$", "$\\begin{pmatrix}4 & 5 & -4\\\\\n-3 & -3 & 2\\\\\n-5 & -3 & 1\\\\\n\\end{pmatrix} \\cdot \\begin{pmatrix}2 & 3 & -5\\\\\n-6 & -1 & -4\\\\\n6 & 5 & 3\\\\\n\\end{pmatrix} = \\begin{pmatrix}-46 & -13 & -52\\\\\n24 & 4 & 33\\\\\n14 & -7 & 40\\\\\n\\end{pmatrix}$"]],
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