Question

Junte-se a um colega e estabeleçam três matrizes quadradas, A, B e C, de ordem 2, e verifiquem numericamente a validade das propriedades:

A) (A.B).C = A . (B.C)

B) (A+B).C = A . C+B.C

C) A . L2 = L2 . A = A

D) (K . A) . B = K . (A.B), com k e R

E) (A . B)T = BT . AT

Answer 1

Olá

Vamos considerar as matrizes A = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ , B = $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ e C = $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$

a) (A.B).C = A.(B.C)
(A.B).C = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}1&0\\3&0\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}0&2\\0&6\\\end{array}\right]$

A.(B.C) = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\0&0\\\end{array}\right]$ = $\left[\begin{array}{cc}0&2\\0&6\\\end{array}\right]$

Logo, (A.B).C = A.(B.C)

b) (A+B).C = AC + BC

(A+B).C = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ + $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}2&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}2&6\\4&10\\\end{array}\right]$

A.C + B.C = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ + $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ * $\left[\begin{array}{cc}0&2\\1&1\\\end{array}\right]$ = $\left[\begin{array}{cc}2&4\\4&10\\\end{array}\right]$ + $\left[\begin{array}{cc}0&2\\0&0\\\end{array}\right]$ = $\left[\begin{array}{cc}2&6\\4&10\\\end{array}\right]$

Logo. (A + B).C = A.C + B.C

c) A.L2 = L2.A = A

A.L2 = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}1&0\\0&1\\\end{array}\right]$ = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$

L2.A = $\left[\begin{array}{cc}1&0\\0&1\\\end{array}\right]$ * $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ = $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$

Logo, A.L2 = L2.A = A

d) (K.A).B = K(A.B) 

(K.A).B = K * $\left[\begin{array}{cc}1&2\\3&4\\\end{array}\right]$ * $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ = $\left[\begin{array}{cc}K&2K\\3K&4K\\\end{array}\right]$ * $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ = $\left[\begin{array}{cc}K&0\\3K&0\\\end{array}\right]$

K(A.B) = K * $\left[\begin{array}{cc}1&0\\3&0\\\end{array}\right]$ = $\left[\begin{array}{cc}K&0\\3K&0\\\end{array}\right]$

Logo, (K.A).B = K.(A.B)

e) $( A.B)^{T} = B^{T} . A^{T}$

$( A.B)^{T}$ = $\left[\begin{array}{cc}1&3\\0&0\\\end{array}\right]$ 

$B^{T} . A^{T}$= $\left[\begin{array}{cc}1&0\\0&0\\\end{array}\right]$ * $\left[\begin{array}{cc}1&3\\2&4\\\end{array}\right]$ = $\left[\begin{array}{cc}1&3\\0&0\\\end{array}\right]$.

Logo $( A.B)^{T} = B^{T} . A^{T}$

Obs: atenção nos parênteses