Question

01. Determine o produto entre as matrizes a seguir:

02. Considerando duas matrizes A e B, sendo A = e B = verifique se A x B = B x A.


03. Dadas as matrizes A = B = e M = A. B. Determine a soma dos elementos da matriz M.

04. Elevar um número ao quadrado é o mesmo que multiplicá-lo por ele mesmo duas vezes. Assim, se A = quanto será A²?

Answer 1

$\boxed{\begin{array}{c}\rm{O~produto~de~matrizes~existe}\\\rm{se~e~somente~se}\\\bf{o~n\acute{u}mero~de~colunas~da~1^{\underline{a}}matriz}\\\bf{for~igual~ao~n\acute{u}mero~de~linhas~da~2^{\underline{a}}~matriz}\end{array}}$

$\large\boxed{\begin{array}{c}\tt{01.}~\frak{\underline{Determine~o~produto~~entre~~as~~~matrizes}}\\\frak{\underline{a~~seguir}:}\end{array}}$

$\rm{a)}\begin{bmatrix}2&-2\\2&1\end{bmatrix}\times\begin{bmatrix}1&-1&3\\3&0&2\end{bmatrix}\\=\begin{bmatrix}2\cdot1-2\cdot3&2\cdot(-1)-2\cdot0&2\cdot3-2\cdot2\\2\cdot1+1\cdot3&2\cdot(-1)+1\cdot0&2\cdot3+1\cdot2\end{bmatrix}\\\begin{bmatrix}-4&-2&2\\5&-2&8\end{bmatrix}$

$\rm{b)}~~\begin{bmatrix}\frac{1}{2}&3\\-1&5\end{bmatrix}\times\begin{bmatrix}3&2\\4&5\end{bmatrix}\\\begin{bmatrix}\frac{1}{2}\cdot3+3\cdot4&\frac{1}{2}\cdot2+3\cdot5\\-1\cdot3+5\cdot4&-1\cdot2+5\cdot5\end{bmatrix}\\=\begin{bmatrix}\frac{27}{2}&16\\17&23\end{bmatrix}$

$\boxed{\begin{array}{c}\tt{02.}~~\frak{\underline{Considerando~duas~matrizes~A~e~B,sendo}}\\\sf{A}=\begin{bmatrix}1&2\\7&0\end{bmatrix}~~\sf{e}~~\sf{B}=\begin{bmatrix}1&-1\\0&~~1\end{bmatrix},\\\frak{verifique~se~A\times B=B\times A.}\end{array}}$

$\sf{\underline{Soluc_{\!\!,}\tilde{a}o:}}\\\sf{A\times B}=\begin{bmatrix}1&2\\7&0\end{bmatrix}\times\begin{bmatrix}1&-1\\0&~~1\end{bmatrix}=\begin{bmatrix}1\cdot1+2\cdot0&1\cdot(-1)+2\cdot1\\7\cdot1+0\cdot0&7\cdot(-1)+0\cdot1\end{bmatrix}\\\\\sf{A\times B}=\begin{bmatrix}1&1\\7&-7\end{bmatrix}$

$\sf{B\times A}=\begin{bmatrix}1&-1\\0&~~1\end{bmatrix}\times\begin{bmatrix}1&2\\7&0\end{bmatrix}=\begin{bmatrix}1\cdot1-1\cdot7&1\cdot2-1\cdot0\\0\cdot1+1\cdot7&0\cdot2+1\cdot0\end{bmatrix}\\\sf{B\times A}=\begin{bmatrix}-6&2\\7&0\end{bmatrix}$

$\Large\boxed{\begin{array}{c}\frak{Portanto~~A\times B\ne B\times A}\end{array}}$

$\boxed{\begin{array}{c}\tt{03.}~~\frak{\underline{Dada~as~matrizes~}}\\\sf{A}=\begin{bmatrix}-1&-2\\~~~3&~~1\end{bmatrix},\sf{B}=\begin{bmatrix}~~0&1\\-1&6\end{bmatrix}~~\sf{e}~~\sf{M}=A\cdot B.\\\frak{Determine~a~soma~dos~elementos~da~matriz}~~\sf{M.}\end{array}}$

$\sf{\underline{soluc_{\!\!,}\tilde{a}o}:}\\\sf{M}=\begin{bmatrix}-1&-2\\~~3&~~1\end{bmatrix}\times\begin{bmatrix}~~0&1\\-1&6\end{bmatrix}=\begin{bmatrix}-1\cdot0-2\cdot(-1)&-1\cdot1-2\cdot6\\3\cdot0+1\cdot(-1)&3\cdot1+1\cdot6\end{bmatrix}\\\sf{M}=\begin{bmatrix}~~2&-9\\-1&~~9\end{bmatrix}\\\sf{\sum=2+(-9)+(-1)+9}\\\sf{\sum=2-\diagup\!\!\!\!9-1+\diagup\!\!\!\!9=1}$

$\boxed{\begin{array}{c}\tt{04.}~~\frak{Elevar~um~n\acute{u}mero~ao~quadrado~\acute{e}~o~mesmo}\\\frak{que~multiplic\acute{a}-lo~por~ele~mesmo~duas~vezes.}\\\frak{Assim,se}\\\sf{A}=\begin{bmatrix}-1&3\\0&2\end{bmatrix},~~\frak{quanto~ser\acute{a}}~~\sf{A^2}~?\end{array}}$

$\sf{\underline{Soluc_{\!\!,}\tilde{a}o}:}\\\sf{A^2}=\begin{bmatrix}-1&3\\~~0&2\end{bmatrix}\times\begin{bmatrix}-1&3\\~~0&2\end{bmatrix}=\begin{bmatrix}(-1)\cdot(-1)+3\cdot0&-1\cdot3+3\cdot2\\0\cdot(-1)+2\cdot0&0\cdot3+2\cdot2\end{bmatrix}\\\sf{A^2}=\begin{bmatrix}1&3\\0&4\end{bmatrix}$

$\huge\boxed{\begin{array}{c}\tt{Espero~~ter~~ajudado}\\\tt{\acute{O}timos~~estudos~~:)}\end{array}}$