Question

junte-se a um colega e estabeleçam quatro matrizes A, B, C e 0, de mesma ordem e verifiquem numericamente a validade das propriedades

a)A+B= B+A
b)(A+B)+C=A+(B+)
c)A+0=A
d)A+(-A)=0

Answer 1

Vamos estabelecer matriz de valores arbitrários, de ordem 2x2.

$A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] \\ \\ \\ B = \left[\begin{array}{cc}e&f\\g&h\end{array}\right] \\ \\ \\ C = \left[\begin{array}{cc}i&j\\k&l\end{array}\right] \\ \\ \\ 0 = \left[\begin{array}{cc}0&0\\0&0\end{array}\right]$

A operação pedida no enunciado é a de soma. Para somar duas matrizes de mesma ordem, basta somar elemento por elemento, respeitando suas posições na matriz. Como a soma não possui nenhuma ordem específica, podemos dizer que x + y = y + x

Letra A
A + B = B + A

$A+B = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] +\left[\begin{array}{cc}e&f\\g&h\end{array}\right] = \left[\begin{array}{cc}a+e&b+f\\c+g&d+h\end{array}\right] \\ \\ \\ B + A = \left[\begin{array}{cc}e&f\\g&h\end{array}\right] + \left[\begin{array}{cc}a&b\\c&d\end{array}\right] = \left[\begin{array}{cc}e+a&f+b\\g+c&h+d\end{array}\right] \\ \\ \\ \boxed{A+B = B+A}$

Letra B
(A+B)+C = A+(B+C)

$A+B = \left[\begin{array}{cc}a+e&b+f\\c+g&d+h\end{array}\right] \\ \\ \\ (A+B)+C = \left[\begin{array}{cc}a+e&b+f\\c+g&d+h\end{array}\right] + \left[\begin{array}{cc}i&j\\k&l\end{array}\right] \\ \\ \\ (A+B)+C =\left[\begin{array}{cc}a+e+i&b+f+j\\c+g+k&d+h+l\end{array}\right]$

$B+C = \left[\begin{array}{cc}e+i&f+j\\g+k&h+l\end{array}\right] \\ \\ \\ A+(B+C) = \left[\begin{array}{cc}a&b\\c&d\end{array}\right]+ \left[\begin{array}{cc}e+i&f+j\\g+k&h+l\end{array}\right]\\ \\ \\ A+(B+C) = \left[\begin{array}{cc}a+e+i&b+f+j\\c+g+k&d+h+l\end{array}\right] \\ \\ \\ \boxed{(A+B)+C = A+(B+C)}$

Letra C
A + 0 = A

$A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] \\ \\ \\ 0 = \left[\begin{array}{cc}0&0\\0&0\end{array}\right] \\ \\ \\ A+0 = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] + \left[\begin{array}{cc}0&0\\0&0\end{array}\right] = \left[\begin{array}{cc}a+0&b+0\\c+0&d+0\end{array}\right] = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] \\ \\ \\ \boxed{A+0=A}$


Letra D
A + (-A) = 0

$A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] \\ \\ \\ (-A) = \left[\begin{array}{cc}-a&-b\\-c&-d\end{array}\right] \\ \\ \\ A+(-A) = \left[\begin{array}{cc}a&b\\c&d\end{array}\right] + \left[\begin{array}{cc}-a&-b\\-c&-d\end{array}\right] =\left[\begin{array}{cc}a-a&b-b\\c-c&d-d\end{array}\right] \\ \\ \\ A+(-A) = \left[\begin{array}{cc}0&0\\0&0\end{array}\right] \\ \\ \\ \boxed{A+(-A)=0}$