Question

Resolver as seguintes equações:

Answer 1

Resposta:

A soma de matrizes é efetuada da seguinte forma,

$\left[\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right] +\left[\begin{array}{ccc}b_{11}&b_{12}\\b_{21}&b_{22}\end{array}\right]=\left[\begin{array}{ccc}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\end{array}\right]$

Ou seja, na letra a), quando temos,

$\left[\begin{array}{ccc}1&3\\-2&2\end{array}\right] +\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]=\left[\begin{array}{ccc}5&7\\-5&9\end{array}\right]$

Isto significa que,

$\left[\begin{array}{ccc}1&3\\-2&2\end{array}\right] +\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]=\left[\begin{array}{ccc}1+a&3+b\\-2+c&2+d\end{array}\right]=\left[\begin{array}{ccc}5&7\\-5&9\end{array}\right]$

Ora, se

$\left[\begin{array}{ccc}1+a&3+b\\-2+c&2+d\end{array}\right]=\left[\begin{array}{ccc}5&7\\-5&9\end{array}\right]$

Então,

$\left \{ \begin{matrix} 1+a=5\\3+b=7\\-2+c=-5\\2+d=9\end{matrix} \right.$

Vamos isolar cada incógnita e assim descobriremos seus valores,

$1+a=5\\a=5-1\\a=4$

$3+b=7\\b=7-3\\b=4$

$-2+c=-5\\c=-5+2\\c=-3$

$2+d=9\\d=9-2\\d=7$

Encontramos os valores,

$\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]=\left[\begin{array}{ccc}4&4\\-3&7\end{array}\right]$

Na letra b), utilizaremos o mesmo raciocínio.

$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] + \left[\begin{array}{ccc}1&1&1\\0&1&1\\0&0&1\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\1&1&0\\2&1&1\end{array}\right]$

$\left[\begin{array}{ccc}a+1&b+1&c+1\\d+0&e+1&f+1\\g+0&h+0&i+1\end{array}\right]=\left[\begin{array}{ccc}1&0&0\\1&1&0\\2&1&1\end{array}\right]$

Logo,

$\left \{ \begin{matrix} a+1=1\ \rightarrow \ a=1-1=0\\b+1=0 \rightarrow \ b=0-1=-1\\c+1=0 \rightarrow \ c=0-1=-1\\d+0=1 \rightarrow \ d=1-0=1\\e+1=1 \rightarrow \ e=1-1=0\\f+1=0 \rightarrow \ f=0-1=-1\\g+0=2 \rightarrow \ g=2-0=2\\h+0=1 \rightarrow \ h=1-0=1\\i+1=1 \rightarrow \ i=1-1=0 \end{matrix} \right.$

Então a matriz,

$\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] =\left[\begin{array}{ccc}0&-1&-1\\1&0&-1\\2&1&0\end{array}\right]$