Question

Calcule o determinante da matriz 3x3 , cujos elementos são: {aij= i+2j, se i≥j {aij= i²-j, se i

Answer 1

Explicação passo-a-passo:

$\sf A=\left(\begin{array}{ccc} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{array}\right)$

• $\sf a_{11}=1+2\cdot1~\rightarrow~a_{11}=1+2~\rightarrow~a_{11}=3$

• $\sf a_{12}=1^2-2~\rightarrow~a_{12}=1-2~\rightarrow~a_{12}=-1$

• $\sf a_{13}=1^2-3~\rightarrow~a_{13}=1-3~\rightarrow~a_{13}=-2$

• $\sf a_{21}=2+2\cdot1~\rightarrow~a_{21}=2+2~\rightarrow~a_{21}=4$

• $\sf a_{22}=2+2\cdot2~\rightarrow~a_{22}=2+4~\rightarrow~a_{22}=6$

• $\sf a_{23}=2^2-3~\rightarrow~a_{23}=4-3~\rightarrow~a_{23}=1$

• $\sf a_{31}=3+2\cdot1~\rightarrow~a_{31}=3+2~\rightarrow~a_{31}=5$

• $\sf a_{32}=3+2\cdot2~\rightarrow~a_{32}=3+4~\rightarrow~a_{32}=7$

• $\sf a_{33}=3+2\cdot3~\rightarrow~a_{33}=3+6~\rightarrow~a_{33}=9$

Assim:

$\sf A=\left(\begin{array}{ccc} 3&-1&-2 \\ 4&6&1 \\ 5&7&9 \end{array}\right)$

$\sf det~(A)=3\cdot6\cdot9+(-1)\cdot1\cdot5+(-2)\cdot4\cdot7-5\cdot6\cdot(-2)-7\cdot1\cdot3-9\cdot4\cdot(-1)$

$\sf det~(A)=162-5-56+60-21+36$

$\sf det~(A)=258-82$

$\sf det~(A)=176$