Question

supondo as matrizes A,B,C,D e X quadradas de mesma ordem e inversíveis, resolver as equações matriciais nas quais X é a variavel. Com todos os passos :

a) ADX=ABC

b)DXt= DC 

c) ABCX²D²=ABCXD

d) D-¹XD=AC

e) CX= 2B=3B  

desde já agradeço!

Answer 1

Olá, Anelizemendes.

$a) ADX=ABC \Rightarrow A^{-1}ADX=A^{-1}ABC \Rightarrow \\\\ IDX=IBC\Rightarrow DX=BC\Rightarrow D^{-1}DX=D^{-1}BC\Rightarrow\\\\ IX=D^{-1}BC\Rightarrow \boxed{X=D^{-1}BC}$


$b)DX^t= DC \Rightarrow D^{-1}DX^t= D^{-1}DC \Rightarrow IX^t= IC \Rightarrow \\\\ X^t=C\Rightarrow (X^t)^t=C^t\Rightarrow \boxed{X=C^t}$


$c) ABCX\²D\²=ABCXD\Rightarrow X^2D^2=XD\Rightarrow\\\\ XXDD=XD\Rightarrow \underbrace{X^{-1}X}_{=I}XD\underbrace{DD^{-1}}_{=I}=\underbrace{X^{-1}X}_{=I}\underbrace{DD^{-1}}_{=I}\Rightarrow \\\\ IXDI=II\Rightarrow XD=I\Rightarrow \boxed{X=D^{-1}}$


$d) D^{-1}XD=AC\Rightarrow DD^{-1}XDD^{-1}=DACD^{-1}\Rightarrow\\\\ IXI=DACD^{-1}\Rightarrow \boxed{X=DACD^{-1}}$


$e) CX= 2B+3B \Rightarrow CX=5B\Rightarrow C^{-1}CX=C^{-1}\cdot5B\Rightarrow \\\\ \boxed{X=5C^{-1}B}$