Question

Calcular determinante 4x4 -2 -3 - 1 -2 -1 0 1 -2 -3 -1 -4 1 -2 2 -3 -1

 

Como calcular essa determinante 4x4 usando laplace?

Answer 1

O teorema consiste em escolher uma fila (linha,i, ou coluna,j) e multiplicar cada um de seus elementos pelo respectivo cofator:

Vamos pergar a 1ª Coluna:

$<var>M=\left[\begin{array}{cccc}2& -3 &- 1& -2\\ -1& 0& 1& -2 \\-3& -1& -4 &1\\ -2 &2& -3& -1\end{array}\right]</var>$

 

Formula do cofator:

$<var>A_{ij}=(-1)^{1+j}.D_{ij}\\onde;\\D_{ij}=M-Linha_i-Coluna_j</var>$

 

$<var>A_{11}=(-1)^{1+1}.D_{11}\\A_{11}=(-1)^2.\left[\begin{array}{ccc} 0& 1& -2 \\ -1& -4 &1\\ 2& -3& -1\end{array}\right]=1*(-21)=-21</var>$

$<var>A_{21}=(-1)^{2+1}.D_{21}=\\A_{21}=(-1)^3.=\left[\begin{array}{cccc} -3 &- 1& -2\\ -1& -4 &1\\ 2& -3& -1\end{array}\right]=-1.(-44)=44</var>$

$<var>A_{31}=(-1)^{3+1}.D_{31}\\A_{31}=(-1)^4.\left[\begin{array}{cccc} -3 &- 1& -2\\ 0& 1& -2 \\2& -3& -1\end{array}\right]=1.29=29</var>$

$<var>A_{41}=(-1)^[4+1}.D_{41}\\A_{41}=(-1)^5.\left[\begin{array}{cccc} -3 &- 1& -2\\ 0& 1& -2 \\-1& -4 &1\end{array}\right]=-1.17=-17</var>$

 

Entao:

 

$<var>detM=a_{11}.A_{11}+a_{21}.A_{21}+a_{31}.A_{31}+a_{41}.A_{41}=\\ detM=2*(-21)+(-1)*44+(-3)*29+(-2)*(-17)\\ detM=-42-44-87+34= \\detM=-139</var>$