Discussion
For the matrix
\[ M = \begin{bmatrix} 2 & 3+2i & -4 \\ 3-2i & 5 & 6i \\ -4 & -6i & 3 \end{bmatrix}, \]
which of the following statements are correct?
P : \(M\) is skew-Hermitian and \(iM\) is Hermitian
Q : \(M\) is Hermitian and \(iM\) is skew Hermitian
R : eigenvalues of \(M\) are real
S : eigenvalues of \(iM\) are real
Answer: B
The correct statements are Q and R only.
\(M\) equals its conjugate transpose: the diagonal entries 2, 5, 3 are real, and the off-diagonal pairs \(3 + 2i\) and \(3 - 2i\), \(-4\) and \(-4\), \(6i\) and \(-6i\) are conjugates. So \(M\) is Hermitian, and then \((iM)^{*} = -iM^{*} = -iM\), so \(iM\) is skew-Hermitian (Q).
A Hermitian matrix has real eigenvalues (R). The eigenvalues of \(iM\) are \(i\) times those of \(M\), so they are purely imaginary, and real only if they are 0. So S could hold only if every eigenvalue of \(M\) were 0, which for a Hermitian matrix means \(M = 0\); \(M\) is not zero, so S fails. P is false because \(M\) is not skew-Hermitian: its diagonal is not purely imaginary.
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