MCQ 1
Given Hermitian matrices \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C}),\) their product \(\mathbf{A}\mathbf{B}\) is Hermitian if and only if they commute, i.e. \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.\)
- True
- False
MCQ 2
Suppose \(\mathbf{A}\in M_{n,n}(\mathbb{C})\) is skew-Hermitian, i.e. \(\overline{\mathbf{A}}^T=-\mathbf{A},\) then \(\mathbf{A}^2\) is skew-Hermitian.
- True
- False
MCQ 3
Suppose \(\mathbf{A}\in M_{n,n}(\mathbb{C})\) is skew-Hermitian, i.e. \(\overline{\mathbf{A}}^T=-\mathbf{A},\) then \(\mathbf{A}^3\) is skew-Hermitian.
- True
- False
MCQ 4
Suppose \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) are Hermitian, then \(\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}\) is skew-Hermitian.
- True
- False
MCQ 5
Suppose \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) are skew-Hermitian, then \(\mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}\) is skew-Hermitian.
- True
- False
MCQ 6
Suppose \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) are Hermitian, then \(\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}\) is Hermitian.
- True
- False
MCQ 7
Suppose \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) are skew-Hermitian, then \(\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}\) is Hermitian.
- True
- False
MCQ 8
Suppose \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) such that one is Hermitian and the other is skew-Hermitian, then \(\mathbf{A}\mathbf{B}+\mathbf{B}\mathbf{A}\) is skew-Hermitian.
- True
- False
MCQ 9
Suppose \(A,B\in M_{n,n}(\mathbb{C})\) such that one is Hermitian and the other is skew-Hermitian, then \(AB-BA\) is Hermitian.
- True
- False
MCQ 10
The space of Hermitian matrices forms a complex vector space.
- True
- False