Question

If \(A=\begin{bmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{bmatrix}\), the value of \(|\text{Adj}(A^{2026})|\) is equal to:

• A. \(1\)
• B. \(-1\)
• C. \(-\frac{1}{2}\)
• D. \(\frac{1}{2}\)

Hint:

The matrix is a rotation by 120 degrees, so its powers repeat every three steps and its determinant is 1.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The matrix has the form \(\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\) with \(\theta=120^{\circ}\), a rotation matrix.

Step 2: Find the power
\(A^{n}\) is the rotation by \(n\theta\). Since \(A^{3}\) is rotation by \(360^{\circ}\), \(A^{3}=I\).
\(2026=3\times 675+1\), so \(A^{2026}=A\).

Step 3: Determinant of the adjoint
For a 2 by 2 matrix B, \(|\text{Adj}\,B|=|B|^{2-1}=|B|\).
\(|A|=\frac{1}{4}+\frac{3}{4}=1\).

Step 4: Result
So \(|\text{Adj}(A^{2026})|=|A|=1\).

Final Answer:
The required value is 1. \[ \boxed{\text{(A) }1} \]