Question

This transformation is called a rotation with shear. Can you see why it might be called that? Let's investigate more. Find the determinant of the rotation matrix. Det r = ______, which matches the determinant for our other ______. Find the product of the matrix and its transpose: r·rt = ______. Reflection and rotation matrices.

Answer 1

The determinant of a rotation matrix is +1 or -1, which indicates it preserves volume or area. This characteristic is shared with reflection matrices. The product of a rotation matrix and its transpose results in the identity matrix, emphasizing that rotation matrices are orthogonal.

Step-by-step explanation:

The student is asking about matrix operations pertaining to rotational transformations and their properties in linear algebra, which is a subject of mathematics. When we discuss a rotation with shear, we typically refer to a composition of transformations including rotation and shear operations. A pure rotation matrix has some distinct properties, such as having a determinant of 1 or -1, indicating that it preserves area (in 2D) or volume (in 3D) while a shear matrix may have a different determinant.

To answer the student's question, for a rotation matrix R, the determinant (Det R) would be +1 or -1, depending on the direction of the rotation. This determinant value matches that of other rotation matrices as well as reflection matrices with respect to areas and volumes. The product of a rotation matrix R and its transpose RT is the identity matrix, which indicates that rotation matrices are orthogonal.