Problem-11

system of equations
orthogonal matrices

Question

If \(A\) is an orthogonal matrix in \(\mathbb{R}^{n \times n}\) and \(b \in \mathbb{R}^{n}\), then which of the following statements are true?

  1. \(A\) has orthonormal columns.

  2. \(Ax = b\) has a unique solution.

  3. \(Ax = b\) has no solutions.

  4. \(Ax = b\) has infinitely many solutions.

Solution

Options (a) and (b) are correct.

A \(n \times n\) matrix \(A\) is orthogonal if its columns are orthonormal. It follows that the columns of \(A\) are linearly independent. This implies that \(A\) is invertible and \(Ax = b\) has a unique solution. It is to be noted that the columns of an orthogonal matrix have to be orthonormal and not just orthogonal.