Orthogonal Matrices Quiz

What is the defining property of an orthogonal matrix?

• It is always a diagonal matrix.

• Its transpose is equal to its inverse.

• Its determinant is always 1.

• It has only integer entries.

Which of the following is an example of an orthogonal matrix?

• A matrix representing a counterclockwise rotation.

• A matrix with all zero entries.

• A matrix with all positive entries.

• A matrix with a determinant of zero.

What is the result when an orthogonal matrix is multiplied by its transpose?

• A matrix with negative entries.

• A zero matrix.

• A diagonal matrix.

• The identity matrix.

What does the theorem about orthogonal matrices and orthonormal bases state?

• Orthogonal matrices have both orthonormal rows and columns.

• Orthogonal matrices have orthonormal rows only.

• Orthogonal matrices have orthonormal columns only.

• Orthogonal matrices have no orthonormal properties.

In the proof of the theorem, what is shown about the dot products of row vectors?

• They are always greater than one.

• They are zero for distinct rows and one for identical rows.

• They are always negative.

• They are equal to the determinant of the matrix.

What is the determinant of an orthogonal matrix?

• It is always positive.

• It can be any real number.

• It is always zero.

• It is either 1 or -1.

Which property is preserved by orthogonal transformations?

• The trace of the matrix.

• The rank of the matrix.

• The sum of the matrix entries.

• The norm of vectors.

What happens to the angles between vectors under an orthogonal transformation?

• They are halved.

• They remain unchanged.

• They are doubled.

• They become zero.

What is a key characteristic of the product of two orthogonal matrices?

• It is always a zero matrix.

• It is not necessarily orthogonal.

• It is also an orthogonal matrix.

• It has a determinant of zero.

How does swapping rows in an orthogonal matrix affect its orthogonality?

• It doubles the determinant.

• It does not affect the orthogonality.

• It makes the matrix non-orthogonal.

• It changes the determinant to zero.