Question

Is the set of all orthogonal 3-by-3 matrices with determinant= -1 a group? If so, what is the unit element?

Short answer

No, it does not form a group.

Step-by-step solution

Given Information

The set of all orthogonal 3 by 3 matrices with determinants $=-1$.

Orthogonal Matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors in linear algebra.

Orthogonal Matrix

The set of all orthogonal 3 by 3 matrices with determinants$=-1$ is not a group. The unit element with respect to matrix multiplication is the unit matrix, which has determinant.

Therefore, it is not a part of this set of matrices, which means that this set does not have a unit element and cannot be a group.

Thus,$3\times 3$ orthogonal matrices with determinants -1 do not form a group.