Question

Ache os valores de m, n p e q de modo que:

Answer 1

A soma, subtração e igualdade entre matrizes são operações feitas entre elementos de mesma posição, sendo assim, temos:

$\left[\begin{array}{ccc}m&2m\\p&p\end{array}\right]~+~\left[\begin{array}{ccc}n&-n\\q&-3q\end{array}\right] ~=~\left[\begin{array}{ccc}7&8\\1&5\end{array}\right]\\\\\\\\\left[\begin{array}{ccc}m+n&2m+(-n)\\p+q&p+(-3q)\end{array}\right]~=~\left[\begin{array}{ccc}7&8\\1&5\end{array}\right]\\\\\\\\\left[\begin{array}{ccc}m+n&2m-n\\p+q&p-3q\end{array}\right]~=~\left[\begin{array}{ccc}7&8\\1&5\end{array}\right]\\\\\\\\\left\{\begin{array}{ccc}m+n&=&7\\2m-n&=&8\\p+q&=&1\\p-3q&=&5\end{array}\right$

Chegamos a 4 equações e 4 incógnitas.

$Somando\!-\!\!se~a~primeira~equacao~com~a~segunda:\\\\\\(m+n)+(2m-n)~=~7+8\\\\\\3m~=~15\\\\\\m~=~\frac{15}{3}\\\\\\\boxed{m~=~5}\\\\\\Substituindo~o~valor~de~"m"~na~primeira~(ou~segunda)~equacao:\\\\\\m+n~=~7\\\\\\5+n~=~7\\\\\\\boxed{n~=~2}$

$Somando\!-\!\!se~a~terceira~equacao~com~o~negativo~da~quarta:\\\\\\(p+q)~-~(p-3q)~=~1-5\\\\\\p+q-p+3q~=~-4\\\\\\4q~=~-4\\\\\\\boxed{q~=\,-1}\\\\\\Substituindo~o~valor~de~"q"~na~terceira~(ou~quarta)~equacao:\\\\\\p+q~=~1\\\\\\p+(-1)~=~1\\\\\\p~=~1+1\\\\\\\boxed{p~=~2}$