Animated Solution for Mathematics - Matrices and Determinants: Let $f(x)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|,x\in [0,\pi ]$. Then the maximum value of $f(x)$ is equal to

The determinant provided is a $3\times 3$ matrix containing terms of ${\sin }^{2}x$, ${\cos }^{2}x$, and $\cos 2x$. At first glance, direct expansion appears to be an algebraic nightmare.

To avoid this, we employ the strategy of row and column operations. Whenever you see repeating patterns or similar terms across rows or columns, simplifying the structure before expansion is the most efficient path.

Row Operations

Let us focus on the first two rows. Notice that the first column has ${\sin }^{2}x$ in row one and $2+{\sin }^{2}x$ in row two. By applying the operation $R_1\to R_1-R_2$, the ${\sin }^{2}x$ terms cancel out.

The first row becomes:

Next, we simplify the relationship between row two and row three using $R_2\to R_2-R_3$. This operation yields:

We can now write the simplified determinant as:

By using these two row operations, we have eliminated almost all complex trigonometric terms from the first two rows. Expanding this determinant is now a straightforward process.

We expand along the first row ($R_1$) to take advantage of the zero:

Evaluating the $2\times 2$ minors carefully, we obtain:

For the first minor:

For the second minor:

Substituting these back into our expansion:

Distributing the constants, we get:

Grouping the terms:

Recalling the fundamental trigonometric identity $\cos 2x={\cos }^{2}x-{\sin }^{2}x$, we substitute this into the expression:

This simplifies to the compact function:

We are asked to find the maximum value of $f(x)$ in the interval $[0,\pi ]$. To maximize $f(x)=4+2\cos 2x$, we must maximize the term $\cos 2x$.

The maximum value of $\cos 2x$ is $1$, which occurs at $x=0$ and $x=\pi$ within the given interval. Substituting this value:

The maximum value of the original determinant is $6$.