Question

Let $A = \begin{bmatrix} 5 & \sin^2\theta & \cos^2\theta \\ -\sin^2\theta & -5 & 1 \\ \cos^2\theta & 1 & 5 \end{bmatrix}$, then the maximum value of $\det(A)$ is:

• A. $-125$
• B. $200$
• C. $\dfrac{255}{2}$
• D. $145$

Hint:

Use test values like $\theta = 0$ and $\theta = \pi/2$ to evaluate max/min if range isn’t complex.

Answer 1

The Correct Option is A

This is a determinant dependent on $\theta$, but to find its \emph{maximum} value, we substitute suitable values like $\theta = 0$ or $\theta = \pi/2$. Try $\theta = 0$: $\sin^2\theta = 0$, $\cos^2\theta = 1$
Then $A = \begin{bmatrix} 5 & 0 & 1 \\ 0 & -5 & 1\\ 1 & 1 & 5 \end{bmatrix}$ Computing determinant: \[ \det(A) = 5(-5 \cdot 5 - 1 \cdot 1) - 0 + 1(0 \cdot 1 - (-5) \cdot 1) = 5(-25 - 1) + 5 = -130 + 5 = -125 \]