Let T be an orthogonal operator on ℝ2 and let A be its matrix representation with respect to the standard ordered basis of ℝ2. Which one of the following st

To answer the question regarding the orthogonal operator $T$ on $\mathbb{R}^2$ and its matrix representation $A$, we need to consider the properties of orthogonal matrices.

Properties of Orthogonal Matrices

1. Definition: A matrix $A$ is called orthogonal if its columns (and rows) are orthonormal vectors. This translates mathematically to the condition: $A^TA = I$ where $I$ is the identity matrix and $A^T$ is the transpose of $A$.

2. Determinant: The determinant of an orthogonal matrix is either $1$ or $-1$: $\text{det}(A) = \pm 1$

3. Matrix Representation: In $\mathbb{R}^2$, any orthogonal operator can be represented by a matrix of the form: $A = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$ for a rotation by an angle $\theta$, or a reflection matrix.

Conclusion for Given Options

Without the specific options that follow the question, we cannot definitively select one. However, it is clear that if asked which properties an orthogonal operator or its matrix representation must satisfy, we can conclude that:

1. The matrix must satisfy $A^TA = I$.
2. The determinant must be $\pm 1$.
3. It may represent a rotation or reflection in $\mathbb{R}^2$.

If you could provide the specific options, I can help identify the correct one based on these properties.