The maximum value of the determinant of the matrix $\left[\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \ \sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right] \text { is }$

Question

The maximum value of the determinant of the matrix $\left[\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \ \sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right] \text { is }$, 0, 2, 4, 6

MARKS App · Accepted Answer

Given, $\left|\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 2 x \ \sin ^2 x & 1+\cos ^2 x & 4 \sin 2 x \ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right|$ Applying $R_1 \rightarrow R_1-R_3$ and $R_2 \rightarrow R_2-R_3$, we get $=\left|\begin{array}{ccc} 1 & 0 & -1 \ 0 & 1 & -1 \ \sin ^2 x & \cos ^2 x & 1+4 \sin 2 x \end{array}\right|$ Applying $C_2 \rightarrow C_2+C_1$ $=\left|\begin{array}{ccc} 1 & 1 & -1 \ 0 & 1 & -1 \ \sin ^2 x & 1 & 1+4 \sin 2 x \end{array}\right|$ Applying $R_2 \rightarrow R_2-R_1$ and $R_3 \rightarrow R_3-R_1$ $=\left|\begin{array}{ccc} 1 & 1 & -1 \ -1 & 0 & 0 \ -1+\sin ^2 x & 0 & 2+4 \sin 2 x \end{array}\right|=2+4 \sin 2 x$ Since, maximum value of $\sin 2 x$ is 1. $=2+4=6$

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