Every orthogonal matrix is normal.

• True
• False

If \(\mathbf R\in \mathrm O(n)\) is orthogonally diagonalisable, i.e. there is a matrix \(\mathbf U\in \mathrm O(n)\) such that \(\mathbf U\mathbf R \mathbf U^T\) is diagonal, then \(\mathbf R\) must be symmetric.

• True
• False

The matrix \(\begin{psmallmatrix}a & b\\ 5 & b\end{psmallmatrix}\) cannot be unitary, for any value of \(x.\)

• True
• False

The matrix \(\begin{psmallmatrix}x & \frac{1}{2}\\ -\frac{1}{2} & x\end{psmallmatrix}\) can only be unitary for value of \(x=\frac{\sqrt{3}}{2}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*=\mathbf U.\) Then \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*=\mathbf U.\) Then \(\mathbf U^2=\mathbf{1}_{n}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^2=\mathbf{1}_{n}.\) Then \(\mathbf U^*=\mathbf U.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^2=\mathbf{1}_{n}.\) Then \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\) Then \(\mathbf U^*=\mathbf U.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\) Then \(\mathbf U^2=\mathbf{1}_{n}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*\mathbf U=\mathbf{1}_{n}\) and \(\mathbf U^2=\mathbf{1}_{n}.\) Then \(\mathbf U^*=\mathbf U.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^2=\mathbf{1}_{n}\) and \(\mathbf U^*=\mathbf U.\) Then \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\)

• True
• False

Suppose \(\mathbf U\in M_{n,n}(\mathbb{C})\) satisfies \(\mathbf U^*=\mathbf U\) and \(\mathbf U^*\mathbf U=\mathbf{1}_{n}.\) Then \(\mathbf U^2=\mathbf{1}_{n}.\)

• True
• False

If a unitary matrix is triangular then it is diagonal.

• True
• False