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<Text-field style="Heading 1" layout="Normal">determinant</Text-field>
<Text-field style="Text" layout="Normal"> (abrev. <Font bold="true">det</Font>), <Font italic="true">n</Font><Font encoding="UTF-8">. de uma matriz quadrada A \303\251 uma certa soma alternada de produtos das entradas de A, uma de cada linha e coluna. Determinantes t\303\252m a propriedade que </Font></Text-field>
<Text-field style="Text" layout="Normal">det(AB) = det(A) x det(B).</Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">Se A \303\251 a matriz 2 x 2</Font></Text-field>
<Text-field style="2D Math" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
<Text-field style="Text" layout="Normal"><Font encoding="UTF-8">det(A) \303\251 igual a </Font><Font italic="true">ad - bc</Font>. O determinante de uma matriz de ordem <Font italic="true">n</Font><Font encoding="UTF-8"> pode ser avaliado como uma certa combina\303\247\303\243o linear dos dedeterminantes de matrizes de ordem </Font><Font italic="true">n</Font> - 1 (por exemplo, seus <Hyperlink linktarget="Dictionary:Definition/cofactor" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">cofatores</Font></Hyperlink>).  Por exemplo, o determinante de uma 3 x 3 </Text-field>
<Text-field style="2D Math" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY4LUkjbWlHRiQ2JVEkZGV0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiQtRjY2KC1GIzYmLUYsNiNRIUYnLUknbXRhYmxlR0YkNjctSSRtdHJHRiQ2KC1JJG10ZEdGJDYoLUYsNiVRImFGJ0YvRjIvJSlyb3dhbGlnbkdGQC8lLGNvbHVtbmFsaWduR0ZALyUrZ3JvdXBhbGlnbkdGQC8lKHJvd3NwYW5HUSIxRicvJStjb2x1bW5zcGFuR0ZVLUZINigtRiw2JVEiYkYnRi9GMkZNRk9GUUZTRlYtRkg2KC1GLDYlUSJjRidGL0YyRk1GT0ZRRlNGVkZNRk9GUS1GRTYoLUZINigtRiw2JVEiZEYnRi9GMkZNRk9GUUZTRlYtRkg2KC1GLDYlUSJlRidGL0YyRk1GT0ZRRlNGVi1GSDYoLUYsNiVRImZGJ0YvRjJGTUZPRlFGU0ZWRk1GT0ZRLUZFNigtRkg2KC1GLDYlUSJnRidGL0YyRk1GT0ZRRlNGVi1GSDYoLUYsNiVRImhGJ0YvRjJGTUZPRlFGU0ZWLUZINigtRiw2JVEiaUYnRi9GMkZNRk9GUUZTRlZGTUZPRlEvJSZhbGlnbkdRJWF4aXNGJy9GTlEpYmFzZWxpbmVGJy9GUFEnY2VudGVyRicvRlJRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0YxLyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGW3IvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZnIvJSZmcmFtZUdGZnIvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR1EmZmFsc2VGJy8lLWVxdWFsY29sdW1uc0dGYHMvJS1kaXNwbGF5c3R5bGVHRmBzLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmNyRj4vRjNRJ25vcm1hbEYnRmpzL0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGXHRGanNGanMtSSNtb0dGJDYtUSI9RidGanMvJSZmZW5jZUdGYHMvJSpzZXBhcmF0b3JHRmBzLyUpc3RyZXRjaHlHRmBzLyUqc3ltbWV0cmljR0Zgcy8lKGxhcmdlb3BHRmBzLyUubW92YWJsZWxpbWl0c0dGYHMvJSdhY2NlbnRHRmBzLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaXVGSi1GZnQ2LVEoJiM4OTAxO0YnRmpzRml0Rlt1Rl11Rl91RmF1RmN1RmV1L0ZodVEmMC4wZW1GJy9GW3ZGYHZGKy1GNjYkLUYjNiQtRjY2KC1GIzYmRj4tRkI2Ni1GRTYnRmNvRmhvRk1GT0ZRLUZFNidGZHBGaXBGTUZPRlFGXnFGYXFGY3FGZXFGZ3FGaXFGXHJGXnJGYXJGZHJGZ3JGaXJGW3NGXnNGYXNGY3NGZXNGaHNGPkZqc0Zqc0ZcdEZfdEZidEZcdEZqc0Zqcy1GZnQ2LVEifkYnRmpzRml0Rlt1Rl11Rl91RmF1RmN1RmV1Rl92RmF2LUZmdDYtUSgmIzg3MjI7RidGanNGaXRGW3VGXXVGX3VGYXVGY3VGZXVGX3ZGYXZGYHdGWkZcdkYrLUY2NiQtRiM2JC1GNjYoLUYjNiZGPi1GQjY2LUZFNidGXm9GaG9GTUZPRlEtRkU2J0ZfcEZpcEZNRk9GUUZecUZhcUZjcUZlcUZncUZpcUZcckZeckZhckZkckZnckZpckZbc0Zec0Zhc0Zjc0Zlc0Zoc0Y+RmpzRmpzRlx0Rl90RmJ0Rlx0RmpzRmpzRmB3LUZmdDYtUSIrRidGanNGaXRGW3VGXXVGX3VGYXVGY3VGZXUvRmh1USwwLjIyMjIyMjJlbUYnL0ZbdkZoeEZgd0ZpbkZcdkYrLUY2NiQtRiM2JC1GNjYoLUYjNiZGPi1GQjY2LUZFNidGXm9GY29GTUZPRlEtRkU2J0ZfcEZkcEZNRk9GUUZecUZhcUZjcUZlcUZncUZpcUZcckZeckZhckZkckZnckZpckZbc0Zec0Zhc0Zjc0Zlc0Zoc0Y+RmpzRmpzRlx0Rl90RmJ0Rlx0RmpzRmpzRmpz">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</Equation></Text-field>
<Text-field style="Text" layout="Normal">Formalmente, se A = [<Font italic="true">a<Font subscript="true">ij</Font></Font><Font encoding="UTF-8">]  \303\251 uma matriz </Font><Font italic="true">n</Font> <Font family="SansSerif">x</Font> <Font italic="true">n</Font> com entradas em um <Hyperlink linktarget="Dictionary:Definition/ring" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">anel</Font></Hyperlink> <Hyperlink linktarget="Dictionary:Definition/unital" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman" encoding="UTF-8">unit\303\241rio</Font></Hyperlink><Font encoding="UTF-8"> comutativo, ent\303\243o 
</Font>  det (A) = <Equation executable="false" style="2D Math" input-equation="Sigma[`in`(sigma, S[n])]">NiMmSSZTaWdtYUc2IjYjLUkjaW5HNiI2JEkmc2lnbWFHNiImSSJTRzYiNiNJIm5HNiI=</Equation><Equation executable="false" style="2D Math" input-equation="epsilon(sigma)">NiMtSShlcHNpbG9uRzYiNiNJJnNpZ21hRzYi</Equation> <Equation executable="false" style="2D Math" input-equation="a[sigma[1]]">NiMmSSJhRzYiNiMqJiIiIiIiIiZJJnNpZ21hRzYiNiMiIiIiIiI=</Equation> <Equation executable="false" style="2D Math" input-equation="a[2*sigma[2]]">NiMmSSJhRzYiNiMqJiIiIyIiIiZJJnNpZ21hRzYiNiMiIiMiIiI=</Equation> ... <Equation executable="false" style="2D Math" input-equation="a[n*sigma[n]]">NiMmSSJhRzYiNiMqJkkibkc2IiIiIiZJJnNpZ21hRzYiNiNJIm5HNiIiIiI=</Equation>, 
onde S<Font subscript="true" italic="true">n</Font><Font encoding="UTF-8"> \303\251 o conjunto de </Font><Hyperlink linktarget="Dictionary:Definition/permutation" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman" encoding="UTF-8">permuta\303\247\303\265es</Font></Hyperlink> dos inteiros de 1 a <Font italic="true">n</Font>, <Font family="MaplePi">e</Font>(<Font family="MaplePi">s</Font><Font encoding="UTF-8">)  \303\251 a </Font><Hyperlink linktarget="Dictionary:Definition/signature" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">assinatura</Font></Hyperlink><Font encoding="UTF-8"> da permuta\303\247\303\243o </Font><Font family="MaplePi">s</Font>, e <Font family="MaplePi">s</Font><Font subscript="true" italic="true">i</Font><Font encoding="UTF-8">  \303\251 o </Font><Font italic="true">i</Font><Font superscript="true" encoding="UTF-8">\303\251simo</Font><Font encoding="UTF-8"> membro da permuta\303\247\303\243o </Font><Font family="MaplePi">s</Font><Font encoding="UTF-8">.  Uma matriz \303\251 invers\303\255vel se e somente se seu determinante \303\251 n\303\243o zero, e os determinantes podem ser usados na solu\303\247\303\243o de equa\303\247\303\265es simult\303\242neas, etc. para m\303\251todos de matriz, embora a </Font><Hyperlink linktarget="Dictionary:Definition/Gaussianelimination" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman" encoding="UTF-8">elimina\303\247\303\243o Gaussiana</Font></Hyperlink><Font encoding="UTF-8"> ou t\303\251cnicas relacionadas s\303\243o quase sempre preferidas em c\303\241lculos atuais para matrizes maiores que 3 </Font><Font family="SansSerif">x</Font> 3. Compare <Hyperlink linktarget="Dictionary:Definition/permanent" hyperlink="true"><Font style="Dictionary Hyperlink" family="Times New Roman">permanente</Font></Hyperlink><Font encoding="UTF-8">. Veja tamb\303\251m </Font><Hyperlink linktarget="Help:LinearAlgebra[Determinant]" hyperlink="true"><Font style="Hyperlink" family="Times New Roman">LinearAlgebra[Determinant]</Font></Hyperlink>, <Hyperlink linktarget="Help:Student[LinearAlgebra][Determinant]" hyperlink="true"><Font style="Hyperlink" family="Times New Roman">Student[LinearAlgebra][Determinant]</Font></Hyperlink>.</Text-field>
</Worksheet>