Question

Calcule o Determinante:
D:
| 3 1 -2 1 |

| 5 2 2 3 |

| 7 4 -5 0 |

| 1 -1 11 2 |

Answer 1

Resposta:

$D =108$

Explicação passo-a-passo:

Vamos lá!  Aplicaremos o Teorema de Laplace nesse pequeno probleminha.

• Escolheremos a última coluna por que tem um zero que facilitará o cálculo.

$\left[\begin{array}{cccc}3&1&-2&1\\5&2&2&3\\7&4&-5&0\\1&-1&11&2\end{array}\right] ||1|3|0|2||$

$D =a_{14}\cdot Cof_{14} +a_{24}\cdot Cof_{24} +a_{34}\cdot Cof_{34} +a_{44}\cdot Cof_{44}$

• Vamos eliminar o cofator de Zero por ser um elemento nulo.

Multiplicando a última coluna pelos seus cofatores, temos:

• Pela Regra de Sarrus, multiplique a diagonal principal pela diagonal secundária...

$a_{14} .(-1)^{1+4} \cdot Cof_{14}$

$1\cdot(-1)^{1+4} \cdot\left[\begin{array}{ccc}5&2&2\\7&4&-5\\1&-1&11\end{array}\right] \\\\D=(-1)^{5} \cdot\{[220-10-14]-[8+25+154]\}\\\\D=(-1)^{5} \cdot[196-187]\\\\D =1\cdot(-1)\cdot9\\\\D=-9\\\\\\a_{24} .(-1)^{2+4} \cdot Cof_{24}\\\\3\cdot(-1)^{2+4} \cdot\left[\begin{array}{ccc}3&1&-2\\7&4&-5\\1&-1&11\end{array}\right]\\\\D =1\cdot\{[132-5+14]-[-8+15+77]\}\\\\ D =1\cdot[141-84]\\\\D =3\cdot1\cdot57\\\\D =171$

$a_{44}\cdot(-1)^{4+4}\cdot Cof_{44} \\\\2\cdot(-1)^{4+4} \cdot\left[\begin{array}{ccc}3&1&-2\\5&2&2\\7&4&-5\end{array}\right]\\\\D=1\cdot\{[-30+14-40]-[-28+24-25]\}=\\\\D=1\cdot[-56-(-29)]=\\\\D=1\cdot[-56+29]\\\\D=2\cdot1\cdot[-27]=-54\\\\\\D=-9+171-54\\\\D =-9+117\\\\D =108$

Bons estudos!!!