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3 and 4 .Determinants and Matrices
normal
Let $A =$ $\left[ {\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ - \sin \theta }&1&{\sin \theta }\\{ - 1}&{ - \sin \theta }&1\end{array}} \right]$, where $0 \le \theta < 2\pi$ , then
A
$Det\, (A) = 0$
B
$Det\, A \in (0, \infty )$
C
$Det\, (A) \in [2, 4]$
D
$Det\, A \in [2, \infty )$
Solution
$| A | =$ $\left| {\,\begin{array}{*{20}{c}}1&{\sin \theta }&1\\{ – \sin \theta }&1&{\sin \theta }\\{ – 1}&{ – \sin \theta }&1\end{array}\,} \right|$
$= 1(1 + \sin^2\theta ) – \sin\theta (- \sin\theta + \sin\theta ) + (1 + \sin^2\theta ) = 2\, (1 + \sin^2\theta )$ $| \sin\theta | \le 1$
$==> -1 \le \sin\theta \le 1$
$==> 0 \le \sin^2\theta \le 1$
==> $1 \le 1 + \sin^2\theta \le 2$
$==> 2 \le 2(1 + \sin2\theta ) \le 4$
==> $| A | \in [2, 4]$
Standard 12
Mathematics
