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

$$A=7548$$,$$B=4375$$ and $$C=-537-4$$ $$BC=4375-537-4=1001=I$$ $$(BC)k=I$$ ∑$$k=0$$∞$$13kTr{A(BC)k}=Tr(A)+13Tr(A)+132Tr(A)+$$.....∞ ⇒∑$$k=0$$∞$$13kTr{A(BC)k}=Tr(A)1+13+132+$$.....$$=(7+8)11-13=452$$

Questions

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 $\sum_{k = 0}^{\infty} \frac{1}{3^{k}} T r {A {(B C)}^{k}} =$

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detailed solution

Correct option is A

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