Question

Verifique as relações da matriz 3x3, algebricamente:

a) tr ([B]+[A]) = tr ([A]+ tr [B])
b) tr (C[a]) = C tr [A]
c) tr ([A][b]) = tr ([B][A])
d) tr ([A]-¹[B][a]) = tr ([b])

Answer 1

c) 
$\displaystyle \text{Sea }A=\ \textless \ a_{i,j}\ \textgreater \ \text{ con }1\leq i\leq 3\text{ y }1\leq j \leq 3\text{ y }\\ B=\ \textless \ b_{j,k}\ \textgreater \ \text{ con }1\leq j\leq 3\text{ y }1\leq k \leq 3\text{ y }\\ \\ AB=\ \textless \ AB_{i.k}\ \textgreater \ =\sum\limits_{j=1}^{3}a_{i,j}b_{j,k}\\ \\ tr(AB) = \sum\limits_{h=1}^3 \ \textless \ AB_{h,h}\ \textgreater \ =\sum\limits_{h=1}^3\sum\limits_{j=1}^{3}a_{h,j}b_{j,h}\\ \\ tr(BA) = \sum\limits_{h=1}^3 \ \textless \ BA_{h,h}\ \textgreater \ =\sum\limits_{h=1}^3\sum\limits_{j=1}^{3}b_{h,j}a_{j,h}\\ \\ tr(BA) = \sum\limits_{h=1}^3\sum\limits_{j=1}^{3}a_{j,h}b_{h,j}$

entonces $tr(AB) = tr (BA)$

d) Utilice (c)

$tr(A^{-1}BA)=tr(A^{-1}[BA])=tr([BA]A^{-1})=tr(B[AA^{-1}])=tr(BI)\\ \\ tr(A^{-1}BA)=tr (B)$