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V. Matricer – 3. Lineære ligninger 416 • Multiplikation af række med tal r = 0 1 0 a11 a12 = 0 r a21 a22 a11 a12 ra21 ra22 • Addition af et multiplum af en række til en anden 1 r a11 a12 a11 + ra21 a12 + ra22 = 0 1 a21 a22 3.28. Ensidig invers er tosidig ☞ [LA] 7 Rækkeoperations-matricer Sætning 10 Lad A,B vœre kvadratiske matricer af samme størelse. Så gœlder ”En højre invers er også en venstre invers.” a21 AB = I ⇔ BA = I Bevis Hvis AB = I har alle ligningssystemer Ax = b en løsning x = Bb. Den reducerede form af A kan da ikke have en 0-række og er derfor enhedsmatricen I. Altså findes en matrix C så CA = I. Til slut er C = C(AB) = (CA)B = B. 3.29. Invers ved operationer ☞ [LA] 7 Rækkeoperations-matricer Sætning a22
V. Matricer – 3. Lineære ligninger 417 En kvadratisk matrix A er invertibel, hvis og kun hvis dens reducerede form er enhedsmatricen I. I så fald er den augmenterede matrix (A|I) ∼ (I|A −1 ) ”Den inverse matrix beregnes ved rækkeoperationer på den augmenterede matrix.” 3.30. Invers 2x2-matrix Eksempel 4 ☞ [LA] 6 Løsningsteknik Løs matrixligningen, i.e. beregn invers, 2 1 x11 5 3 x21 x12 1 = x22 0 0 1 2 5 1 1 0 2 1 1 0 ∼ 1 3 0 1 0 2 −5 2 ∼ 2 1 0 2 0 6 −2 1 0 3 ∼ ∼ 0 1 −5 2 0 1 −5 1 1 1 −5 −1 2 0 2 3.31. Invers 2x2-matrix ☞ [LA] 6 Løsningsteknik Eksempel 4 - fortsat
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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