If A = [cos x, sin x, −sin x, cos x], find x satisfying 0 < x < π/2 when A + Aₜ = I.

To solve for \( x \) in the equation \( A = [\cos x, \sin x, -\sin x, \cos x] \) under the condition \( 0 \), we need to analyze the components of the vector \( A \).

Understanding the Components

The vector \( A \) consists of four elements:

• \(\cos x\)
• \(\sin x\)
• \(-\sin x\)
• \(\cos x\)

Setting the Condition

Since you mentioned finding \( x \) satisfying \( 0 \), we can interpret this as needing to find values of \( x \) where the components of \( A \) equal zero.

Finding Values of \( x \)

To satisfy \( A = 0 \), both \(\cos x\) and \(\sin x\) must be zero. This occurs at:

• \( \cos x = 0 \) when \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \)
• \( \sin x = 0 \) when \( x = n\pi \) for any integer \( n \)

Conclusion

Thus, there are no values of \( x \) that can satisfy both conditions simultaneously. Therefore, \( A \) cannot equal zero for any real \( x \).