Let A=7548, B=4375 and C=-537-4 . If Tr(S) denotes the trace of a square matrix S then ∑k=0∞13kTr{A(BC)k}=

A
45/2
B
36
C
81/2
D
9

Solution:

$A = [\begin{matrix} 7 & 5 \\ 4 & 8 \end{matrix}], B = [\begin{matrix} 4 & 3 \\ 7 & 5 \end{matrix}] a n d C = [\begin{matrix} - 5 & 3 \\ 7 & - 4 \end{matrix}] B C = [\begin{matrix} 4 & 3 \\ 7 & 5 \end{matrix}] [\begin{matrix} - 5 & 3 \\ 7 & - 4 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = I {(B C)}^{k} = I \sum_{k = 0}^{\infty} \frac{1}{3^{k}} T r {A {(B C)}^{k}} = T r (A) + \frac{1}{3} T r (A) + \frac{1}{3^{2}} T r (A) + . . . . . \infty \Rightarrow \sum_{k = 0}^{\infty} \frac{1}{3^{k}} T r {A {(B C)}^{k}} = T r (A) (1 + \frac{1}{3} + \frac{1}{3^{2}} + . . . . .) = (7 + 8) (\frac{1}{1 - \frac{1}{3}}) = \frac{45}{2}$

Let $A = [\begin{matrix} 7 & 5 \\ 4 & 8 \end{matrix}], B = [\begin{matrix} 4 & 3 \\ 7 & 5 \end{matrix}]$ and $C = [\begin{matrix} - 5 & 3 \\ 7 & - 4 \end{matrix}]$ . If Tr(S) denotes the trace of a square matrix S then $\sum_{k = 0}^{\infty} \frac{1}{3^{k}} T r {A {(B C)}^{k}} =$

Solution:

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