Question

Let each of the following matrices represent an active transformation of vectors in the (x, \(y\) ) plane (axes fixed, vectors rotated or reflected). As in Example \(3,\) show that each matrix is orthogonal, find its determinant, and find the rotation angle, or find the line of reflection. $$\left(\begin{array}{rr}0 & -1 \\\\-1 & 0\end{array}\right)$$.

Short answer

The given matrix is orthogonal with a determinant -0f and it represents a reflection across the line y = - x.

Step-by-step solution

- Verify Orthogonality

A matrix is orthogonal if its transpose is equal to its inverse. For the given matrix you need to compute both the transpose and inverse.First, compute the transpose of the matrix To find the inverse of the matrix Calculate the determinant for this matrix.

- Check Transpose Equal Inverse

Compare the transpose with the inverse. Since they are equal, the matrix is orthogonal.

- Find the Determinant

Recall that the determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is det \(\begin{pmatrix} 0 & -1 \ -1 & 0\ \end{pmatrix}\) = \( (0)(0) - (-1)(-1) = -1 \).

- Identify the Transformation

Given that the determinant is this matrix represents a reflection. To find the line of reflection, note that this is the result of reflecting vectors across the line \( y = - x \)

Key concepts

Matrix Transformations

Matrix transformations are a way to manipulate vectors in a plane using matrices. Each matrix represents a specific type of transformation, such as rotation or reflection. The given matrix is \(\begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}\), and it's important to distinguish the type of transformation it represents.
You must verify whether the matrix is orthogonal before analyzing the transformation type.
An orthogonal matrix must satisfy the condition that its transpose is equal to its inverse.
Once the matrix is confirmed to be orthogonal, you can further determine its impact, whether it causes rotation or reflection.
To check for orthogonality:

• Calculate the transpose of the given matrix.
• Compute the inverse of the matrix.
• Compare if the transpose equals the inverse, proving orthogonality.

This step establishes the basis of how matrix transformations operate in the (x,y) plane.

Determinants

The determinant is a special value computed from a square matrix that provides important information about the matrix transformation.
For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as: \[ det = (a \times d) - (b \times c) \] The given matrix is \(\begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}\).
Computing the determinant involves substituting the values:

• \((0) \times (0) - (-1) \times (-1) = -1 \)

The determinant helps determine if the transformation is a reflection or rotation.
A determinant of -1 indicates the matrix corresponds to a reflection transformation.
Moreover, the determinant helps in finding if the transformation preserves the area (determinant = ±1).

Reflections and Rotations

Reflections and rotations are common types of transformations represented by matrices. Let's explore how the given matrix corresponds to these transformations:
For the matrix \(\begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix}\), the determinant = -1 suggests it is a reflection.
To identify the precise line of reflection, analyze the effect on vectors.
This specific matrix reflects vectors across the line y = -x.
Here’s how:

• Upon applying the matrix transformation to any vector, the resulting vector is effectively a mirror image across the y = -x line.
• Therefore, this matrix is a reflection matrix for the line y = -x.

This concept is crucial in understanding geometric transformations and their implications on the vector space.
Remember, for a matrix to represent a rotation, its determinant would be 1 and it would rotate vectors by a specific angle. However, with a determinant of -1, we identify reflections along specific lines.