If \(A = \begin{bmatrix}\alpha & 3\\ 3 & \alpha\end{bmatrix}\) and \(|A^{3}| = 343\), then the value of \(\alpha\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\pm 4\)
By the product rule, \(|A^{3}| = |A|^{3}\).
\(|A|^{3} = 343\)
\(|A| = 7\)
(343 has only one real cube root.) Now
\(|A| = \alpha^{2} - 9\)
So
\(\alpha^{2} - 9 = 7\)
\(\alpha^{2} = 16\)
\(\alpha = \pm 4\)
Both values work, since \(\alpha\) appears only as \(\alpha^{2}\).
Why the other options are incorrect
Option (2) 4 only — Incorrect. \(\alpha = -4\) also gives \(|A| = 16 - 9 = 7\).
Option (3) \(-4\) only — Incorrect. \(\alpha = 4\) also gives \(|A| = 7\).
Option (4) \(\pm\sqrt{2}\) — Incorrect. This comes from \(|A| = -7\), but \((-7)^{3} = -343\), not 343.