Question-13

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?

• \(A\) has orthogonal columns which are not necessarily orthonormal.
• \(A\) has orthonormal columns.
• \(Ax = b\) has a unique solution.
• \(Ax = b\) has no solutions.
• \(Ax = b\) has infinitely many solutions.

Answer

• \(A\) has orthogonal columns which are not necessarily orthonormal.
• \(A\) has orthonormal columns.
• \(Ax = b\) has a unique solution.
• \(Ax = b\) has no solutions.
• \(Ax = b\) has infinitely many solutions.

Solution

An \(n \times n\) 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 have to be orthonormal and not just orthogonal.