Considere a matriz \begin{equation} M= \left[ \begin{array}{ccc} a_{1,1} & b_{2,1} & c_{3,1} \\ a_{1,2} & b_{2,2} & c_{3,2} \\ a_{1,3} & b_{2,3} & c_{3,3} \end{array} \right], \end{equation} vamos calcular o determinante desta matriz, denotado por \begin{equation} det(M)= \left| \begin{array}{ccc} a_{1,1} & b_{2,1} & c_{3,1} \\ a_{1,2} & b_{2,2} & c_{3,2} \\ a_{1,3} & b_{2,3} & c_{3,3} \end{array} \right|. \end{equation}

Repare que o determinante de uma matriz é denotado por barras corridas.

Cofatores

Vamos considerar um elemento específico da matriz e definir o cofator deste elemento. considerando o elemento \(\color{green}{a_{1,1}}\). Precisamos considerar os elementos que não pertencem a mesma coluna e linha deste elemento. \begin{equation} M= \left[ \begin{array}{ccc} \color{green}{a_{1,1}} & \color{red}{b_{2,1}} & \color{red}{c_{3,1}} \\ \color{red}{a_{1,2}} & \color{blue}{b_{2,2}} & \color{blue}{c_{3,2}} \\ \color{red}{a_{1,3}} & \color{blue}{b_{2,3}} & \color{blue}{c_{3,3}} \end{array} \right], \end{equation} de forma que o cofator deste elemento é \begin{equation} \kappa_{\color{green}{1,1}}= (-1)^{\color{green}{1+1}} \left| \begin{array}{ccc} \color{blue}{b_{2,2}} & \color{blue}{c_{3,2}} \\ \color{blue}{b_{2,3}} & \color{blue}{c_{3,3}} \end{array} \right| \\ \kappa_{\color{green}{1,1}}= (-1)^{\color{green}{1+1}} (\color{blue}{b_{2,2}c_{3,3}-c_{3,2}b_{2,3}}).\end{equation}

Ou seja, o cofator envolve o cálculo de um detemninante de ordem inferior.

Determinante

O método de Laplace permite o calculo do determinante de uma matriz através do calculo dos cofatores de toda uma linha ou coluna da matriz. Por exemplo, o determinante de \(M\) em relação a segunda linha é \begin{equation} det(M)= \left| \begin{array}{ccc} a_{1,1} & b_{2,1} & c_{3,1} \\ \color{green}{a_{1,2}} & \color{green}{b_{2,2}} & \color{green}{c_{3,2}} \\ a_{1,3} & b_{2,3} & c_{3,3} \end{array} \right| =\\ =\color{green}{a_{1,2}}\kappa_{1,2} + \color{green}{b_{2,2}} \kappa_{2,2} + \color{green}{c_{3,2}} \kappa_{3,2}. \end{equation}

Cofator \(\kappa_{1,2}\)

Da matriz \begin{equation} M= \left[ \begin{array}{ccc} \color{red}{a_{1,1}} & \color{blue}{b_{2,1}} & \color{blue}{c_{3,1}} \\ \color{green}{a_{1,2}} & \color{red}{b_{2,2}} & \color{red}{c_{3,2}} \\ \color{red}{a_{1,3}} & \color{blue}{b_{2,3}} & \color{blue}{c_{3,3}} \end{array} \right], \end{equation} temos \begin{equation} \kappa_{\color{green}{1,2}}= (-1)^{\color{green}{1+2}} \left| \begin{array}{ccc} \color{blue}{b_{2,1}} & \color{blue}{c_{3,1}} \\ \color{blue}{b_{2,3}} & \color{blue}{c_{3,3}} \end{array} \right| .\end{equation}

Cofator \(\kappa_{2,2}\)

Da matriz \begin{equation} M= \left[ \begin{array}{ccc} \color{blue}{a_{1,1}} & \color{red}{b_{2,1}} & \color{blue}{c_{3,1}} \\ \color{red}{a_{1,2}} & \color{green}{b_{2,2}} & \color{red}{c_{3,2}} \\ \color{blue}{a_{1,3}} & \color{red}{b_{2,3}} & \color{blue}{c_{3,3}} \end{array} \right], \end{equation} temos \begin{equation} \kappa_{\color{green}{2,2}}= (-1)^{\color{green}{2+2}} \left| \begin{array}{ccc} \color{blue}{a_{1,1}} & \color{blue}{c_{3,1}} \\ \color{blue}{a_{1,3}} & \color{blue}{c_{3,3}} \end{array} \right| .\end{equation}

Cofator \(\kappa_{3,2}\)

Da matriz \begin{equation} M= \left[ \begin{array}{ccc} \color{blue}{a_{1,1}} & \color{blue}{b_{2,1}} & \color{red}{c_{3,1}} \\ \color{red}{a_{1,2}} & \color{red}{b_{2,2}} & \color{green}{c_{3,2}} \\ \color{blue}{a_{1,3}} & \color{blue}{b_{2,3}} & \color{red}{c_{3,3}} \end{array} \right], \end{equation} temos \begin{equation} \kappa_{\color{green}{3,2}}= (-1)^{\color{green}{3+2}} \left| \begin{array}{ccc} \color{blue}{a_{1,1}} & \color{blue}{b_{2,1}} \\ \color{blue}{a_{1,3}} & \color{blue}{b_{2,3}} \end{array} \right| .\end{equation}