︠e61d179c-5a8f-46b9-9f57-7efcdcbcf8a9i︠ %html

Linearna algebra

Zadatak 1

Zadajte simboličke matrice $$\mathbf{A}=\left[\begin{array}{cc}a_{11}&a_{12}\\ a_{21} & a_{22}\end{array}\right],\quad \mathbf{B}=\left[\begin{array}{cc}b_{11}&b_{12}\\ b_{21} & b_{22}\end{array}\right],$$ te izračunajte matricu $\mathbf{C}$ definiranu izrazom $$\mathbf{C}=\mathbf{AB}-\mathbf{BA}.$$ Kada bi $\mathbf A$ i $\mathbf B$ bili brojevi a ne matrice, kolika bi bila vrijednost $\mathbf C$?

Matrica je pravokutna tablica brojeva. Matrica reda m x n ima m redaka i n stupaca. Zadana matrica $\mathbf{A}$ je reda 2x2. Vidi ovaj link, ali ne ovaj !

︡22de2f82-19be-4eae-83d6-470e7363fe57︡{"done":true,"html":"

Linearna algebra

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Zadatak 1

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Zadajte simboličke matrice $$\\mathbf{A}=\\left[\\begin{array}{cc}a_{11}&a_{12}\\\\ a_{21} & a_{22}\\end{array}\\right],\\quad \\mathbf{B}=\\left[\\begin{array}{cc}b_{11}&b_{12}\\\\ b_{21} & b_{22}\\end{array}\\right],$$ te izračunajte matricu $\\mathbf{C}$ definiranu izrazom $$\\mathbf{C}=\\mathbf{AB}-\\mathbf{BA}.$$ Kada bi $\\mathbf A$ i $\\mathbf B$ bili brojevi a ne matrice, kolika bi bila vrijednost $\\mathbf C$?

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\nMatrica je pravokutna tablica brojeva. Matrica reda m x n ima m redaka i n stupaca. Zadana matrica $\\mathbf{A}$ je reda 2x2. Vidi ovaj link, ali ne ovaj !\n

"} ︠0a08df62-e38b-4697-91d3-ea41c0a6dcc3s︠ A = matrix(2, 2, var('a11', 'a12', 'a21', 'a22')) show(A) show(A*A) # Matrice se mogu množiti pod određenim uvjetima na njihove redove (detaljnije ćete vidjeti u Matematici 1) i rezultat množenja je opet matrica. Umnožak dvije matrice A i B realiziramo koristeći operator * , isto kao i kod množenja brojeva, A*B # Matrice istog reda se mogu i zbrajati i oduzimati korištenjem standarnih operatora + i - i rezultat ovih operacija je opet matrica B = matrix(2, 2, var('b11', 'b12', 'b21', 'b22')) show(B) C = A*B - B*A show(C) ︡db394c7c-547d-46c8-81cf-3e252b6f5722︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\na_{11} & a_{12} \\\\\na_{21} & a_{22}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rr}\na_{11}^{2} + a_{12} a_{21} & a_{11} a_{12} + a_{12} a_{22} \\\\\na_{11} a_{21} + a_{21} a_{22} & a_{12} a_{21} + a_{22}^{2}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rr}\nb_{11} & b_{12} \\\\\nb_{21} & b_{22}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rr}\n-a_{21} b_{12} + a_{12} b_{21} & -a_{12} b_{11} + a_{11} b_{12} - a_{22} b_{12} + a_{12} b_{22} \\\\\na_{21} b_{11} - a_{11} b_{21} + a_{22} b_{21} - a_{21} b_{22} & a_{21} b_{12} - a_{12} b_{21}\n\\end{array}\\right)$
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Zadatak 2

Unesite dvije matrice, $\mathbf A$ i $\mathbf B$, čiji matrični elementi su brojevi, koje su obje kvadratne, regularne i reda barem 3x3. Izračunajte:

Ovdje $\mathbf{A}^{-1}$ označava matricu inverznu matrici $\mathbf A$, a $\det(\mathbf A)$ označava determinantu matrice $\mathbf A$. Determinanta matrice je broj. Više o tome u Matematici 1.

Matrica je kvadratna ako ima jednak broj redaka i stupaca. Znači kvadratne matrice su uvijek reda m x m.

U trenutku rješavanja ovog labosa još vjerojatno ne znate za pojam regularna matrica. Bit će dovoljno da unesete neke dvije matrice $\mathbf A$ i $\mathbf B$, takve da su im determinante različite od nule. Vidjet ćete kasnije u Matematici 1 da je to upravo jedna od karakterizacija regularnih matrica.

︡245fa92f-eed2-4f1b-a02a-cc4d02f4302b︡{"done":true,"html":"

Zadatak 2

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Unesite dvije matrice, $\\mathbf A$ i $\\mathbf B$, čiji matrični elementi su brojevi, koje su obje kvadratne, regularne i reda barem 3x3. Izračunajte:

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Ovdje $\\mathbf{A}^{-1}$ označava matricu inverznu matrici $\\mathbf A$, a $\\det(\\mathbf A)$ označava determinantu matrice $\\mathbf A$. Determinanta matrice je broj. Više o tome u Matematici 1.

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Matrica je kvadratna ako ima jednak broj redaka i stupaca. Znači kvadratne matrice su uvijek reda m x m.

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U trenutku rješavanja ovog labosa još vjerojatno ne znate za pojam regularna matrica. Bit će dovoljno da unesete neke dvije matrice $\\mathbf A$ i $\\mathbf B$, takve da su im determinante različite od nule. Vidjet ćete kasnije u Matematici 1 da je to upravo jedna od karakterizacija regularnih matrica.

"} ︠79e9228f-46b5-4b89-ace6-80026e19aa42s︠ ex = matrix(2, 2, [1, 2, 0, 1]) # Definirali smo matricu A sa konkretnim brojevima. A je reda 2x2. show(ex) show(ex.det()) # Determinanta matrice A show(ex.inverse()) # Matrica inverzna matrici A A = matrix(3, 3, [1, 6, 3, 9, 1, 4, 0, 2, 0]) B = matrix(3, 3, [1, 5, 3, 4, 1, 2, 0, 9, 2]) show(A) show(B) show(A.det()) show(B.det()) show(A.det() * A.inverse().det()) show((A * B).det() - A.det() * B.det()) ︡f9d16c15-decf-44c4-a010-962920cabde4︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n1 & 2 \\\\\n0 & 1\n\\end{array}\\right)$
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$\\displaystyle 1$
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$\\displaystyle \\left(\\begin{array}{rr}\n1 & -2 \\\\\n0 & 1\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 6 & 3 \\\\\n9 & 1 & 4 \\\\\n0 & 2 & 0\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 5 & 3 \\\\\n4 & 1 & 2 \\\\\n0 & 9 & 2\n\\end{array}\\right)$
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$\\displaystyle 46$
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$\\displaystyle 52$
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$\\displaystyle 1$
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$\\displaystyle 0$
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Zadatak 3

Definirajte simboličke matrice
$$\mathbf A=\left[\matrix{\alpha_{1,1}&\alpha_{1,2}&\alpha_{1,3}\cr\alpha_{2,1}&\alpha_{2,2}&\alpha_{2,3}\cr\alpha_{3,1}&\alpha_{3,2}&\alpha_{3,3}\cr}\right]\quad\hbox{ i }\quad\mathbf B=\left[\matrix{\beta_{1,1}&\beta_{1,2}&\beta_{1,3}&\beta_{1,4}&\beta_{1,5}\cr\beta_{2,1}&\beta_{2,2}&\beta_{2,3}&\beta_{2,4}&\beta_{2,5}\cr\beta_{3,1}&\beta_{3,2}&\beta_{3,3}&\beta_{3,4}&\beta_{3,5}\cr\beta_{4,1}&\beta_{4,2}&\beta_{4,3}&\beta_{4,4}&\beta_{4,5}\cr\beta_{5,1}&\beta_{5,2}&\beta_{5,3}&\beta_{5,4}&\beta_{5,5}\cr}\right].$$

Izračunajte $\det \mathbf B$ i $\mathbf A^{-1}$.

︡88ca23c3-fe8a-4e43-80f6-15b73aa133ec︡{"html": "

Zadatak 3

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Definirajte simboli\u010dke matrice
$$\\mathbf A=\\left[\\matrix{\\alpha_{1,1}&\\alpha_{1,2}&\\alpha_{1,3}\\cr\\alpha_{2,1}&\\alpha_{2,2}&\\alpha_{2,3}\\cr\\alpha_{3,1}&\\alpha_{3,2}&\\alpha_{3,3}\\cr}\\right]\\quad\\hbox{ i }\\quad\\mathbf B=\\left[\\matrix{\\beta_{1,1}&\\beta_{1,2}&\\beta_{1,3}&\\beta_{1,4}&\\beta_{1,5}\\cr\\beta_{2,1}&\\beta_{2,2}&\\beta_{2,3}&\\beta_{2,4}&\\beta_{2,5}\\cr\\beta_{3,1}&\\beta_{3,2}&\\beta_{3,3}&\\beta_{3,4}&\\beta_{3,5}\\cr\\beta_{4,1}&\\beta_{4,2}&\\beta_{4,3}&\\beta_{4,4}&\\beta_{4,5}\\cr\\beta_{5,1}&\\beta_{5,2}&\\beta_{5,3}&\\beta_{5,4}&\\beta_{5,5}\\cr}\\right].$$

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Izra\u010dunajte $\\det \\mathbf B$ i $\\mathbf A^{-1}$.

"}︡ ︠22a1213b-902e-4958-b652-b73cea30e658s︠ def simbolicka_matrica(v, n): xevi = list(var(v + '_%d%d' %(i,j)) for i in interval(1, n) for j in interval(1, n)) return matrix(n, xevi) A = simbolicka_matrica('alpha', 2) show(A) # Da si olakšate, koristite definiranu rutinu simbolicka_matrica(). B = simbolicka_matrica('alpha', 3) C = simbolicka_matrica('beta', 5) show(C.det()) show(B.inverse()) ︡9b281b3b-8bf3-46e6-9358-6399c382e140︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n\\alpha_{11} & \\alpha_{12} \\\\\n\\alpha_{21} & \\alpha_{22}\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\beta_{15} \\beta_{24} \\beta_{33} \\beta_{42} \\beta_{51} - \\beta_{14} \\beta_{25} \\beta_{33} \\beta_{42} \\beta_{51} - \\beta_{15} \\beta_{23} \\beta_{34} \\beta_{42} \\beta_{51} + \\beta_{13} \\beta_{25} \\beta_{34} \\beta_{42} \\beta_{51} + \\beta_{14} \\beta_{23} \\beta_{35} \\beta_{42} \\beta_{51} - \\beta_{13} \\beta_{24} \\beta_{35} \\beta_{42} \\beta_{51} - \\beta_{15} \\beta_{24} \\beta_{32} \\beta_{43} \\beta_{51} + \\beta_{14} \\beta_{25} \\beta_{32} \\beta_{43} \\beta_{51} + \\beta_{15} \\beta_{22} \\beta_{34} \\beta_{43} \\beta_{51} - \\beta_{12} \\beta_{25} \\beta_{34} \\beta_{43} \\beta_{51} - \\beta_{14} \\beta_{22} \\beta_{35} \\beta_{43} \\beta_{51} + \\beta_{12} \\beta_{24} \\beta_{35} \\beta_{43} \\beta_{51} + \\beta_{15} \\beta_{23} \\beta_{32} \\beta_{44} \\beta_{51} - \\beta_{13} \\beta_{25} \\beta_{32} \\beta_{44} \\beta_{51} - \\beta_{15} \\beta_{22} \\beta_{33} \\beta_{44} \\beta_{51} + \\beta_{12} \\beta_{25} \\beta_{33} \\beta_{44} \\beta_{51} + \\beta_{13} \\beta_{22} \\beta_{35} \\beta_{44} \\beta_{51} - \\beta_{12} \\beta_{23} \\beta_{35} \\beta_{44} \\beta_{51} - \\beta_{14} \\beta_{23} \\beta_{32} \\beta_{45} \\beta_{51} + \\beta_{13} \\beta_{24} \\beta_{32} \\beta_{45} \\beta_{51} + \\beta_{14} \\beta_{22} \\beta_{33} \\beta_{45} \\beta_{51} - \\beta_{12} \\beta_{24} \\beta_{33} \\beta_{45} \\beta_{51} - \\beta_{13} \\beta_{22} \\beta_{34} \\beta_{45} \\beta_{51} + \\beta_{12} \\beta_{23} \\beta_{34} \\beta_{45} \\beta_{51} - \\beta_{15} \\beta_{24} \\beta_{33} \\beta_{41} \\beta_{52} + \\beta_{14} \\beta_{25} \\beta_{33} \\beta_{41} \\beta_{52} + \\beta_{15} \\beta_{23} \\beta_{34} \\beta_{41} \\beta_{52} - \\beta_{13} \\beta_{25} \\beta_{34} \\beta_{41} \\beta_{52} - \\beta_{14} \\beta_{23} \\beta_{35} \\beta_{41} \\beta_{52} + \\beta_{13} \\beta_{24} \\beta_{35} \\beta_{41} \\beta_{52} + \\beta_{15} \\beta_{24} \\beta_{31} \\beta_{43} \\beta_{52} - \\beta_{14} \\beta_{25} \\beta_{31} \\beta_{43} \\beta_{52} - \\beta_{15} \\beta_{21} \\beta_{34} \\beta_{43} \\beta_{52} + \\beta_{11} \\beta_{25} \\beta_{34} \\beta_{43} \\beta_{52} + \\beta_{14} \\beta_{21} \\beta_{35} \\beta_{43} \\beta_{52} - \\beta_{11} \\beta_{24} \\beta_{35} \\beta_{43} \\beta_{52} - \\beta_{15} \\beta_{23} \\beta_{31} \\beta_{44} \\beta_{52} + \\beta_{13} \\beta_{25} \\beta_{31} \\beta_{44} \\beta_{52} + \\beta_{15} \\beta_{21} \\beta_{33} \\beta_{44} \\beta_{52} - \\beta_{11} \\beta_{25} \\beta_{33} \\beta_{44} \\beta_{52} - \\beta_{13} \\beta_{21} \\beta_{35} \\beta_{44} \\beta_{52} + \\beta_{11} \\beta_{23} \\beta_{35} \\beta_{44} \\beta_{52} + \\beta_{14} \\beta_{23} \\beta_{31} \\beta_{45} \\beta_{52} - \\beta_{13} \\beta_{24} \\beta_{31} \\beta_{45} \\beta_{52} - \\beta_{14} \\beta_{21} \\beta_{33} \\beta_{45} \\beta_{52} + \\beta_{11} \\beta_{24} \\beta_{33} \\beta_{45} \\beta_{52} + \\beta_{13} \\beta_{21} \\beta_{34} \\beta_{45} \\beta_{52} - \\beta_{11} \\beta_{23} \\beta_{34} \\beta_{45} \\beta_{52} + \\beta_{15} \\beta_{24} \\beta_{32} \\beta_{41} \\beta_{53} - \\beta_{14} \\beta_{25} \\beta_{32} \\beta_{41} \\beta_{53} - \\beta_{15} \\beta_{22} \\beta_{34} \\beta_{41} \\beta_{53} + \\beta_{12} \\beta_{25} \\beta_{34} \\beta_{41} \\beta_{53} + \\beta_{14} \\beta_{22} \\beta_{35} \\beta_{41} \\beta_{53} - \\beta_{12} \\beta_{24} \\beta_{35} \\beta_{41} \\beta_{53} - \\beta_{15} \\beta_{24} \\beta_{31} \\beta_{42} \\beta_{53} + \\beta_{14} \\beta_{25} \\beta_{31} \\beta_{42} \\beta_{53} + \\beta_{15} \\beta_{21} \\beta_{34} \\beta_{42} \\beta_{53} - \\beta_{11} \\beta_{25} \\beta_{34} \\beta_{42} \\beta_{53} - \\beta_{14} \\beta_{21} \\beta_{35} \\beta_{42} \\beta_{53} + \\beta_{11} \\beta_{24} \\beta_{35} \\beta_{42} \\beta_{53} + \\beta_{15} \\beta_{22} \\beta_{31} \\beta_{44} \\beta_{53} - \\beta_{12} \\beta_{25} \\beta_{31} \\beta_{44} \\beta_{53} - \\beta_{15} \\beta_{21} \\beta_{32} \\beta_{44} \\beta_{53} + \\beta_{11} \\beta_{25} \\beta_{32} \\beta_{44} \\beta_{53} + \\beta_{12} \\beta_{21} \\beta_{35} \\beta_{44} \\beta_{53} - \\beta_{11} \\beta_{22} \\beta_{35} \\beta_{44} \\beta_{53} - \\beta_{14} \\beta_{22} \\beta_{31} \\beta_{45} \\beta_{53} + \\beta_{12} \\beta_{24} \\beta_{31} \\beta_{45} \\beta_{53} + \\beta_{14} \\beta_{21} \\beta_{32} \\beta_{45} \\beta_{53} - \\beta_{11} \\beta_{24} \\beta_{32} \\beta_{45} \\beta_{53} - \\beta_{12} \\beta_{21} \\beta_{34} \\beta_{45} \\beta_{53} + \\beta_{11} \\beta_{22} \\beta_{34} \\beta_{45} \\beta_{53} - \\beta_{15} \\beta_{23} \\beta_{32} \\beta_{41} \\beta_{54} + \\beta_{13} \\beta_{25} \\beta_{32} \\beta_{41} \\beta_{54} + \\beta_{15} \\beta_{22} \\beta_{33} \\beta_{41} \\beta_{54} - \\beta_{12} \\beta_{25} \\beta_{33} \\beta_{41} \\beta_{54} - \\beta_{13} \\beta_{22} \\beta_{35} \\beta_{41} \\beta_{54} + \\beta_{12} \\beta_{23} \\beta_{35} \\beta_{41} \\beta_{54} + \\beta_{15} \\beta_{23} \\beta_{31} \\beta_{42} \\beta_{54} - \\beta_{13} \\beta_{25} \\beta_{31} \\beta_{42} \\beta_{54} - \\beta_{15} \\beta_{21} \\beta_{33} \\beta_{42} \\beta_{54} + \\beta_{11} \\beta_{25} \\beta_{33} \\beta_{42} \\beta_{54} + \\beta_{13} \\beta_{21} \\beta_{35} \\beta_{42} \\beta_{54} - \\beta_{11} \\beta_{23} \\beta_{35} \\beta_{42} \\beta_{54} - \\beta_{15} \\beta_{22} \\beta_{31} \\beta_{43} \\beta_{54} + \\beta_{12} \\beta_{25} \\beta_{31} \\beta_{43} \\beta_{54} + \\beta_{15} \\beta_{21} \\beta_{32} \\beta_{43} \\beta_{54} - \\beta_{11} \\beta_{25} \\beta_{32} \\beta_{43} \\beta_{54} - \\beta_{12} \\beta_{21} \\beta_{35} \\beta_{43} \\beta_{54} + \\beta_{11} \\beta_{22} \\beta_{35} \\beta_{43} \\beta_{54} + \\beta_{13} \\beta_{22} \\beta_{31} \\beta_{45} \\beta_{54} - \\beta_{12} \\beta_{23} \\beta_{31} \\beta_{45} \\beta_{54} - \\beta_{13} \\beta_{21} \\beta_{32} \\beta_{45} \\beta_{54} + \\beta_{11} \\beta_{23} \\beta_{32} \\beta_{45} \\beta_{54} + \\beta_{12} \\beta_{21} \\beta_{33} \\beta_{45} \\beta_{54} - \\beta_{11} \\beta_{22} \\beta_{33} \\beta_{45} \\beta_{54} + \\beta_{14} \\beta_{23} \\beta_{32} \\beta_{41} \\beta_{55} - \\beta_{13} \\beta_{24} \\beta_{32} \\beta_{41} \\beta_{55} - \\beta_{14} \\beta_{22} \\beta_{33} \\beta_{41} \\beta_{55} + \\beta_{12} \\beta_{24} \\beta_{33} \\beta_{41} \\beta_{55} + \\beta_{13} \\beta_{22} \\beta_{34} \\beta_{41} \\beta_{55} - \\beta_{12} \\beta_{23} \\beta_{34} \\beta_{41} \\beta_{55} - \\beta_{14} \\beta_{23} \\beta_{31} \\beta_{42} \\beta_{55} + \\beta_{13} \\beta_{24} \\beta_{31} \\beta_{42} \\beta_{55} + \\beta_{14} \\beta_{21} \\beta_{33} \\beta_{42} \\beta_{55} - \\beta_{11} \\beta_{24} \\beta_{33} \\beta_{42} \\beta_{55} - \\beta_{13} \\beta_{21} \\beta_{34} \\beta_{42} \\beta_{55} + \\beta_{11} \\beta_{23} \\beta_{34} \\beta_{42} \\beta_{55} + \\beta_{14} \\beta_{22} \\beta_{31} \\beta_{43} \\beta_{55} - \\beta_{12} \\beta_{24} \\beta_{31} \\beta_{43} \\beta_{55} - \\beta_{14} \\beta_{21} \\beta_{32} \\beta_{43} \\beta_{55} + \\beta_{11} \\beta_{24} \\beta_{32} \\beta_{43} \\beta_{55} + \\beta_{12} \\beta_{21} \\beta_{34} \\beta_{43} \\beta_{55} - \\beta_{11} \\beta_{22} \\beta_{34} \\beta_{43} \\beta_{55} - \\beta_{13} \\beta_{22} \\beta_{31} \\beta_{44} \\beta_{55} + \\beta_{12} \\beta_{23} \\beta_{31} \\beta_{44} \\beta_{55} + \\beta_{13} \\beta_{21} \\beta_{32} \\beta_{44} \\beta_{55} - \\beta_{11} \\beta_{23} \\beta_{32} \\beta_{44} \\beta_{55} - \\beta_{12} \\beta_{21} \\beta_{33} \\beta_{44} \\beta_{55} + \\beta_{11} \\beta_{22} \\beta_{33} \\beta_{44} \\beta_{55}$
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$\\displaystyle \\left(\\begin{array}{rrr}\n-\\frac{{\\left(\\frac{\\alpha_{13}}{\\alpha_{11}} - \\frac{\\alpha_{12} {\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}}\\right)} {\\left(\\frac{\\alpha_{21} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} - \\frac{\\alpha_{31}}{\\alpha_{11}}\\right)}}{\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}} + \\frac{1}{\\alpha_{11}} - \\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}^{2} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} & \\frac{{\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)} {\\left(\\frac{\\alpha_{13}}{\\alpha_{11}} - \\frac{\\alpha_{12} {\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}}\\right)}}{{\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)} {\\left(\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}\\right)}} + \\frac{\\alpha_{12}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} & -\\frac{\\frac{\\alpha_{13}}{\\alpha_{11}} - \\frac{\\alpha_{12} {\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}}}{\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}} \\\\\n-\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{21} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} - \\frac{\\alpha_{31}}{\\alpha_{11}}\\right)}}{{\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)} {\\left(\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}\\right)}} + \\frac{\\alpha_{21}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} & -\\frac{1}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} + \\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{{\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}^{2} {\\left(\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}\\right)}} & -\\frac{\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}}{{\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)} {\\left(\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}\\right)}} \\\\\n\\frac{\\frac{\\alpha_{21} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\alpha_{11} {\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)}} - \\frac{\\alpha_{31}}{\\alpha_{11}}}{\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}} & -\\frac{\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}}{{\\left(\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}\\right)} {\\left(\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}\\right)}} & \\frac{1}{\\frac{{\\left(\\frac{\\alpha_{13} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{23}\\right)} {\\left(\\frac{\\alpha_{12} \\alpha_{31}}{\\alpha_{11}} - \\alpha_{32}\\right)}}{\\frac{\\alpha_{12} \\alpha_{21}}{\\alpha_{11}} - \\alpha_{22}} - \\frac{\\alpha_{13} \\alpha_{31}}{\\alpha_{11}} + \\alpha_{33}}\n\\end{array}\\right)$
"}︡{"done":true}︡ ︠5b197ac1-1faa-464e-99b1-f7d377e04f81i︠ %html

Zadatak 4

Unesite matricu $$\mathbf A=\left[\matrix{{\sqrt2\over 2}&-{\sqrt2\over 2}\cr {\sqrt2\over 2}&{\sqrt2\over 2}}\right].$$

Izračunajte $\mathbf{A}^{2n}$ za $n\in\{1,2,\ldots,10\}$.

︡baf044e0-c396-4169-84bb-22586ae1799c︡{"html": "

Zadatak 4

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Unesite matricu $$\\mathbf A=\\left[\\matrix{{\\sqrt2\\over 2}&-{\\sqrt2\\over 2}\\cr {\\sqrt2\\over 2}&{\\sqrt2\\over 2}}\\right].$$

\n

Izra\u010dunajte $\\mathbf{A}^{2n}$ za $n\\in\\{1,2,\\ldots,10\\}$.

"}︡ ︠929b53ff-1c01-4202-b032-0fe594437a4d︠ ### Napomena: potenciranje matrica se, kao i kod brojeva, radi pomoću operacije ^ A = matrix(2, 2, [(sqrt(2) / 2), (-sqrt(2) / 2), (sqrt(2) / 2), (sqrt(2) / 2)]) for n in range(1, 11): show(A ^ (2*n)) ︡e1fcbab9-0452-45c8-8287-4594d1f2a7fb︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n0 & -1 \\\\\n1 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n-1 & 0 \\\\\n0 & -1\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n0 & 1 \\\\\n-1 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n1 & 0 \\\\\n0 & 1\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n0 & -1 \\\\\n1 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n-1 & 0 \\\\\n0 & -1\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n0 & 1 \\\\\n-1 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n1 & 0 \\\\\n0 & 1\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n0 & -1 \\\\\n1 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n-1 & 0 \\\\\n0 & -1\n\\end{array}\\right)$
"}︡{"done":true}︡ ︠bdf0d731-9d44-425f-87e4-d80afe8d9ad1i︠ %html

Zadatak 5

Unesite neku kvadratnu matricu $\mathbf B$, čiji matrični elementi su brojevi, oblika $2\times 2$. Nacrtajte vektore $\vec x=(1,0)$ i $\vec y=(0,1)$ redom u crvenoj i plavoj tamnoj boji, te vektore $\mathbf B\vec x$ i $\mathbf B\vec y$ u svijetlim tonovima istih boja.

︡c6912df6-26ad-42bf-ba5e-9f48c38ce371︡{"done":true,"html":"

Zadatak 5

\n

Unesite neku kvadratnu matricu $\\mathbf B$, čiji matrični elementi su brojevi, oblika $2\\times 2$. Nacrtajte vektore $\\vec x=(1,0)$ i $\\vec y=(0,1)$ redom u crvenoj i plavoj tamnoj boji, te vektore $\\mathbf B\\vec x$ i $\\mathbf B\\vec y$ u svijetlim tonovima istih boja.

"} ︠b310266c-7f1c-447e-bde0-a61eedd379d8s︠ x = vector([1, 0]) y = vector([0, 1]) vx = arrow((0, 0), x, color=(0.7, 0, 0)) # Vektor na crtežu reprezentiramo strelicom sa početkom u ishodištu koordinatnog sustava i vrhom u točki čije su koordinate određene vektorom. Boja je zadana u (red, green, blue) obliku, kao kombinacija osnovne tri boje. Intenzitet svake od osnovne tri boje zadan je brojem između 0 i 1. vy = arrow((0, 0), y, color=(0, 0, 0.7)) # Primjena matrice na vektor se isto piše kao operacija množenja između matrica. show(vx+vy, aspect_ratio=1) B = matrix(2, 2, [1, 2, 3, 4]) vbx = arrow((0, 0), B*x, color=(1, 0, 0)) vby = arrow((0, 0), B*y, color=(0, 0, 1)) show(vbx+vby, aspect_ratio=1) ︡403d6851-2898-4555-9cd3-4e354f520b9b︡{"file":{"filename":"/home/user/.sage/temp/project-3ada70f2-8cfd-4f33-947a-bc4cd434a227/111/tmp_aM3N3s.svg","show":true,"text":null,"uuid":"f966cec9-1426-4277-9174-2b75ff51b484"},"once":false}︡{"file":{"filename":"/home/user/.sage/temp/project-3ada70f2-8cfd-4f33-947a-bc4cd434a227/111/tmp_Cvfg32.svg","show":true,"text":null,"uuid":"cfacb512-690b-498d-94e3-075f92619ae8"},"once":false}︡{"done":true}︡ ︠4caf0286-d780-4a1b-b667-28f9800ea7abi︠ %html

Primjer

U ovisnosti o (interaktivno zadanim) parametrima $l,m\in[-1,1]$ nacrtajte u prostoru plohu $$x^2+y^2+lz^2=m.$$

︡8e81b925-11b0-48e1-a0ce-be2e7d1d8a7c︡{"html": "

Primjer

\n

U ovisnosti o (interaktivno zadanim) parametrima $l,m\\in[-1,1]$ nacrtajte u prostoru plohu $$x^2+y^2+lz^2=m.$$

"}︡ ︠ea182c3f-9ea5-48ba-88a6-3d55d701a5c8s︠ x,y,z = var('x, y, z') @interact def f(l=(-1,1), m=(-1,1)): P = implicit_plot3d( x^2+y^2+l*z^2==m, (x,-2,2), (y,-2,2), (z,-2,2), color='black') show(P) 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︠a726f0fa-db33-4a00-a5d3-3772c8647f59i︠ %html

Zadatak 6

Nacrtajte u prostoru ravnine zadane formulama $3x+2y+z=0$ (žute boje) i $x-y-z=0$ (zelene boje), te vektore $(3,2,0)$ i $(1,-1,-1)$, crvenom i plavom bojom.

︡a6c246db-98fc-489b-a945-f4c6ce3f5df2︡{"done":true,"html":"

Zadatak 6

\n

Nacrtajte u prostoru ravnine zadane formulama $3x+2y+z=0$ (žute boje) i $x-y-z=0$ (zelene boje), te vektore $(3,2,0)$ i $(1,-1,-1)$, crvenom i plavom bojom.

"} ︠1d3bc894-72b8-4a90-b3e4-990c16278309s︠ x, y, z = var('x, y, z') P1 = implicit_plot3d(3*x + 2*y + z == 0, (x, -2, 2), (y, -2, 2), (z, -2, 2), color='yellow') P2 = implicit_plot3d(x - y - z == 0, (x, -2, 2), (y, -2, 2), (z, -2, 2), color='green') v1 = arrow((0, 0, 0), vector([3, 2, 0]), color='red') v2 = arrow((0, 0, 0), vector([1, -1, -1]), color='blue') show(P1+P2+v1+v2, aspect_ratio=1) ︡76444d2e-b15b-47e8-84ad-887be2763437︡{"file":{"filename":"d8cf3af3-34d7-4690-bf2c-8ae0a02f2a30.sage3d","uuid":"d8cf3af3-34d7-4690-bf2c-8ae0a02f2a30"}}︡{"done":true}︡ ︠d3016e34-d408-4840-a17e-48aed1ef903di︠ %html

Zadatak 7

Unesite neku kvadratnu matricu $\mathbf A$, čiji matrični elementi su brojevi, reda $3\times3$. Za interaktivno zadani realan broj $\lambda\in[-10,10]$ ispišite sljedeće vrijednosti:

Matrica $\mathbf I$ označava specijalnu matricu koju zovemo matrica identiteta. Rang matrice $\mathbf A$ je cijeli nenegativan broj. Više o rangu matrice u Matematici 1.

︡d87cdea2-cbae-4581-91b0-504561c1861c︡{"done":true,"html":"

Zadatak 7

\n

Unesite neku kvadratnu matricu $\\mathbf A$, čiji matrični elementi su brojevi, reda $3\\times3$. Za interaktivno zadani realan broj $\\lambda\\in[-10,10]$ ispišite sljedeće vrijednosti:

\n\n

Matrica $\\mathbf I$ označava specijalnu matricu koju zovemo matrica identiteta. Rang matrice $\\mathbf A$ je cijeli nenegativan broj. Više o rangu matrice u Matematici 1.

"} ︠d62d36d7-7ff6-4c40-aa28-78a4c8c2332bs︠ A = matrix(2, 2, [1, 2, 0, 0]) show(A) show(A.det()) show(A.rank()) # Rang matrice A #Primijetite da se matrica može množiti s realnim brojem. Rezultat je matrica istog reda show(5*A) B = matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9]) I = matrix.identity(3) @interact def funkcija(l=(-10, 10)): R = B - l*I show(R) show(R.det()) show(R.rank()) ︡ef3022e8-4735-4a75-8541-90467c356fa4︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n1 & 2 \\\\\n0 & 0\n\\end{array}\\right)$
"}︡{"html":"
$\\displaystyle 0$
"}︡{"html":"
$\\displaystyle 1$
"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rr}\n5 & 10 \\\\\n0 & 0\n\\end{array}\\right)$
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