Introduç˜ao `a´Algebra Linear com o gnu-Octave - Departamento de ...
Introduç˜ao `a´Algebra Linear com o gnu-Octave - Departamento de ...
Introduç˜ao `a´Algebra Linear com o gnu-Octave - Departamento de ...
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2.4. DETERMINANTES 45<br />
00 11<br />
00 11<br />
00 11<br />
00 11<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00000000000<br />
11111111111<br />
00 11 00000000000<br />
1111111111100<br />
00 11<br />
00 11<br />
Figura 2.1: Esquema do cálculo do <strong>de</strong>terminante <strong>de</strong> matrizes <strong>de</strong> or<strong>de</strong>m 2<br />
Seja agora A =<br />
⎡<br />
⎢<br />
⎣<br />
⎤<br />
a 11 a 12 a 13<br />
⎥<br />
a 21 a 22 a 23<br />
a 31 a 32 a 33<br />
⎦. Recor<strong>de</strong> que S 3 tem 6 elementos. No quadro seguinte,<br />
indicamos, respectivamente, a permutação σ ∈ S 3 , o seu sinal, e o produto a 1σ(1) a 2σ(2) · · · a nσ(n) .<br />
Obtemos, assim,<br />
Permutação σ ∈ S 3 sgn(σ) a 1σ(1) a 2σ(2) · · · a nσ(n)<br />
(1 2 3) +1 a 11 a 22 a 33<br />
(2 3 1) +1 a 12 a 23 a 31<br />
(3 1 2) +1 a 13 a 21 a 32<br />
(1 3 2) −1 a 11 a 23 a 32<br />
(2 1 3) −1 a 12 a 21 a 33<br />
(3 2 1) −1 a 11 a 22 a 31<br />
|A| = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32<br />
−a 11 a 23 a 32 − a 12 a 21 a 33 − a 11 a 22 a 31<br />
Para fácil memorização, po<strong>de</strong>-se recorrer ao esquema apresentado <strong>de</strong> seguida.<br />
000000000000000000000<br />
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11111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
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00000000000000000000000<br />
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000000000000000000000<br />
111111111111111111111<br />
00 11<br />
00 11<br />
00000000000000000000<br />
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00 11<br />
00 11<br />
00000000000000000000<br />
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00000000000000000000000<br />
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111111111111111111111<br />
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00000000000000000000<br />
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000000000000000000000<br />
111111111111111111111<br />
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00000000000000000000000<br />
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111111111111111111111<br />
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111111111111111111111<br />
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111111111111111111111<br />
00000000000000000000<br />
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00 11<br />
00 11<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
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000000000000000000000<br />
111111111111111111111<br />
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00000000000000000000000<br />
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000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00 11 000000000000000000000<br />
111111111111111111111<br />
00 11<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
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00000000000000000000000<br />
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000000000000000000000<br />
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00000000000000000000<br />
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00000000000000000000000<br />
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000000000000000000000<br />
111111111111111111111<br />
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00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00 11 000000000000000000000<br />
111111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00 11<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00 11 000000000000000000000<br />
111111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00 11<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
000000000000000000000<br />
111111111111111111111<br />
00000000000000000000<br />
11111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
00000000000000000000000<br />
11111111111111111111111<br />
Figura 2.2: Esquema do cálculo do <strong>de</strong>terminante <strong>de</strong> matrizes <strong>de</strong> or<strong>de</strong>m 3, ou a Regra <strong>de</strong><br />
Sarrus<br />
2.4.2 Proprieda<strong>de</strong>s<br />
São consequência da <strong>de</strong>finição os resultados que <strong>de</strong> seguida apresentamos, dos quais omitimos<br />
a <strong>de</strong>monstração.<br />
Teorema 2.4.2. Seja A uma matriz quadrada.
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