Question

Seja A= (aij)3x3, com aij = i + j, e B = (bij)3x3, com bij = j - i. Determine a matriz C, tal que C = A.B

Answer 1

Lembrando que "i" representa a linha e "j", a coluna de um elemento da matriz, temos:

$A~=~\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right]\\\\\\\\A~=~\left[\begin{array}{ccc}1+1&1+2&1+3\\2+1&2+2&2+3\\3+1&3+2&3+3\end{array}\right]\\\\\\\\A~=~\left[\begin{array}{ccc}2&3&4\\3&4&5\\4&5&6\end{array}\right]$

$B~=~\left[\begin{array}{ccc}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{array}\right]\\\\\\\\B~=~\left[\begin{array}{ccc}1-1&2-1&3-1\\1-2&2-2&3-2\\1-3&2-3&3-3\end{array}\right]\\\\\\\\B~=~\left[\begin{array}{ccc}0&1&2\\-1&0&1\\-2&-1&0\end{array}\right]$

Agora, calculando a matriz C:

$C~=~\left[\begin{array}{ccc}2&3&4\\3&4&5\\4&5&6\end{array}\right]~.~\left[\begin{array}{ccc}0&1&2\\-1&0&1\\-2&-1&0\end{array}\right]\\\\\\\\C~=~\left[\begin{array}{ccc}2.0+3.(-1)+4.(-2)&2.1+3.0+4.(-1)&2.2+3.1+4.0\\3.0+4.(-1)+5.(-2)&3.1+4.0+5.(-1)&3.2+4.1+5.0\\4.0+5.(-1)+6.(-2)&4.1+5.0+6.(-1)&4.2+5.1+6.0\end{array}\right]$

$C~=~\left[\begin{array}{ccc}0-3-8&2+0-4&4+3+0\\0-4-10&3+0-5&6+4+0\\0-5-12&4+0-6&8+5+0\end{array}\right] \\\\\\\\C~=~\left[\begin{array}{ccc}-11&-2&7\\-14&-2&10\\-17&-2&13\end{array}\right]$