1. Tardigrade
  2. Question
  3. Mathematics
  4. If three matrices A =[2 1 4 1], B =[3 4 2 3] and C =[3 -4 -2 3] . Then t r( A )+ t r (( ABC /2))+ t r(( A ( BC )2/4)) +tr((A(B C)3/8))+ ldots ∞=

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If three matrices $A =\begin{bmatrix}2 & 1 \\ 4 & 1\end{bmatrix}, B =\begin{bmatrix}3 & 4 \\ 2 & 3\end{bmatrix}$ and $C =\begin{bmatrix}3 & -4 \\ -2 & 3\end{bmatrix} .$ Then $t _{r}( A )+ t _{ r }\left(\frac{ ABC }{2}\right)+ t _{r}\left(\frac{ A ( BC )^{2}}{4}\right)$ $+t_{r}\left(\frac{A(B C)^{3}}{8}\right)+\ldots \infty=$

Matrices

Solution:

$BC =\begin{bmatrix}3 & 4 \\ 2 & 3\end{bmatrix}\begin{bmatrix}3 & -4 \\ -2 & 3\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}= I$
$t _{ r }( A )+ t _{ r }\left(\frac{ ABC }{2}\right)+ t _{ r }\left(\frac{ A ( BC )^{2}}{4}\right)$
$+ t _{ r }\left(\frac{ A ( BC )^{3}}{8}\right)+\ldots \ldots+\infty$
$= t _{ r }( A )+ t _{ r }\left(\frac{ A }{2}\right)+ t _{ r }\left(\frac{ A }{4}\right)+\ldots \ldots \infty$
$= t _{ r }( A )+\frac{1}{2} t _{ r }( A )+\frac{1}{2^{2}} t _{ r }( A )+\ldots \ldots \infty$
$=\frac{ t _{ r }( A )}{1-1 / 2}=2 t _{ r }( A )=2(3)=6$