KCET · Maths · Determinants
If \(A=\left[\begin{array}{ll}\mathrm{k} & 2 \\ 2 & \mathrm{k}\end{array}\right]\) and \(\left|\mathrm{A}^3\right|=125\), then the value of k is
- A \(\pm 2\)
- B \(\pm 3\)
- C \(-5\)
- D \(-4\)
Answer & Solution
Correct Answer
(B) \(\pm 3\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & A=\left[\begin{array}{ll}\mathrm{k} & 2 \\ 2 & \mathrm{k}\end{array}\right] \\ & |\mathrm{A}|=\mathrm{k}^2-4 \\ & |\mathrm{~A}|^3=|\mathrm{A}| \cdot|\mathrm{A}| \cdot|\mathrm{A}|=125 \text {, given } \\ & \Rightarrow\left(\mathrm{k}^2-4\right)^3=125 \\ & \Rightarrow \mathrm{k}^2-4=5 \Rightarrow \mathrm{k}^2=9 \\ & \Rightarrow \mathrm{k}= \pm 3\end{aligned}\)
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