Let $\theta \in \left( {0,{\pi \over 2}} \right)$. If the system of linear equations
$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0$
${\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0$
${\cos ^2}\theta x + {\sin ^2}\theta y + (1 + 4\sin 3\,\theta )z = 0$
has a non-trivial solution, then the value of $\theta$ is :
Solution
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\sin 3\,\theta } \cr
} } \right| = 0$$<br><br>${C_1} \to {C_1} + {C_2}$<br><br>$$\left| {\matrix{
2 & {{{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
2 & {1 + {{\sin }^2}\theta } & {4\sin 3\,\theta } \cr
1 & {{{\sin }^2}\theta } & {1 + 4\sin 3\,\theta } \cr
} } \right| = 0$$<br><br>${R_1} \to {R_1} - {R_2},{R_2} \to {R_2} - {R_3}$<br><br>$$\left| {\matrix{
0 & { - 1} & 0 \cr
1 & 1 & { - 1} \cr
1 & {{{\sin }^2}\theta } & {1 + 4\sin 3\,\theta } \cr
} } \right| = 0$$<br><br>or $4\sin 3\theta = - 2$<br><br>$\sin 3\theta = - {1 \over 2}$<br><br>$\theta = {{7\pi } \over {18}}$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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