Question

The maximum value of $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right|$ is ___________.

Answer 1

Let ∆ = $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right|$


$$\begin{gathered} ∆=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right| \\ \mathrm{Applying}{R}_2\to R_2-R_1\mathrm{and}{R}_3\to R_3-R_1 \\ =\left|\begin{array}{ccc}1& 1& 1\\ 1-1& 1+\sin \theta -1& 1-1\\ 1-1& 1-1& 1+\cos \theta -1\end{array}\right| \\ =\left|\begin{array}{ccc}1& 1& 1\\ 0& \sin \theta & 0\\ 0& 0& \cos \theta \end{array}\right| \\ \mathrm{Expanding}\mathrm{along}{C}_1 \\ =1\left(\sin \theta \cos \theta \right) \\ =\frac{2\sin \theta \cos \theta }{2} \\ =\frac{\sin 2\theta }{2} \\ \mathrm{But},\sin 2\theta \le 1 \\ \Rightarrow \frac{\sin 2\theta }{2}\le \frac{1}{2} \end{gathered}$$

Hence, the maximum value of $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+\sin \theta & 1\\ 1& 1& 1+\cos \theta \end{array}\right|$ is $\underline{\frac{1}{2}}$.