TS EAMCET · Maths · Matrices
\(A\) and \(B\) are two \(3 \times 3\) non-singular matrices such that \(\operatorname{adj} A=|A| B\). If \(\operatorname{tr}(x)\) denotes the trace of a square matrix \(X\) and \(C=\left[\begin{array}{ccc}4 & 4 & 7 \\ 3 & -2 & 5 \\ -2 & 3 & 6\end{array}\right]\), then \(\sum_{k=1}^x \operatorname{tr}\left(\frac{1}{3^k}(A B)^k C\right)\) is equal to
- A 12
- B 4
- C 81
- D ∞ (infinite)
Answer & Solution
Correct Answer
(B) 4
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Given, } \operatorname{adj} A=|A| B \\ & \Rightarrow \quad B=\frac{\operatorname{adj} A}{|A|}=A^{-1} \\ & \text { and } \quad \operatorname{tr}(C)=4-2+6=8 \\ & \text { and } \quad(A B)^k=\left(A \cdot A^{-1}\right)^k=(I)^k=I \\ & \text { Now, }…
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