## Differential Equations and Linear Algebra (4th Edition)

Given $$A=\left[ \begin{array}{cccc}{0} & {0} & {0} & {-3} \\ {0} & {0} & {-7} & {-1} \\ {0} & {2} & {6} & {9} \\ {1} & {8} & {-8} & {-9}\end{array} \right]$$ Since, if we have$$A=\left[ \begin{array}{lll}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array} \right]$$ we get \begin{aligned}\operatorname{det}(A)& =\left|\begin{array}{lll}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array}\right| &=a_{11}( a_{22} a_{33}-a_{23} a_{32})-a_{12}( a_{21} a_{33}-a_{23} a_{31})+a_{13}( a_{21} a_{32}-a_{22} a_{31})\\ & =a_{11}( a_{22} a_{33}-a_{23} a_{32})-a_{12}( a_{21} a_{33}-a_{23} a_{31})+a_{13}( a_{21} a_{32}-a_{22} a_{31})\\ &=a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}-a_{11} a_{23} a_{32}-a_{12} a_{21} a_{33}-a_{13} a_{22} a_{31}\\ \end{aligned} Therefore, we get \begin{aligned} \operatorname{det}(A) &=\left|\begin{array}{cccc}{0} & {0} & {0} & {-3} \\ {0} & {0} & {-7} & {-1} \\ {0} & {2} & {6} & {9} \\ {1} & {8} & {-8} & {-9}\end{array}\right| \\ &=-3 \cdot\left|\begin{array}{ccc}{0} & {0} & {-7} \\ {0} & {2} & {6} \\ {1} & {8} & {-8}\end{array}\right| \\ &=-3 \cdot(0 \cdot 2 \cdot(-8)+0 \cdot 6 \cdot 1+(-7) \cdot 0 \cdot 8-0 \cdot 6 \cdot 8-0 \cdot 0 \cdot(-8)-(-7) \cdot 2 \cdot 1) \\ &=-3 \cdot(-0+0-0-0+0+14)=-3 \cdot 24=42 \end{aligned}