Question

If $z = \begin{vmatrix} -5 & 3+4i & 5-7i \\ 3-4i & 6 & 8+7i \\ 5+7i & 8-7i & 9 \end{vmatrix}$, then z is

• A. Purely real
• B. Purely imaginary
• C. $a + ib$, where $a \neq 0, b \neq 0$
• D. $a + ib$, where $b = 4$

Answer 1

Answer: (A)

Purely real

Answer 2

The given matrix

$$A=\begin{pmatrix} -5 & 3+4i & 5-7i \\ 3-4i & 6 & 8+7i \\ 5+7i & 8-7i & 9 \end{pmatrix}.$$

is a Hermitian matrix because:

• $a_{12} = 3+4i$ and $a_{21} = 3-4i$
• $a_{13} = 5-7i$ and $a_{31} = 5+7i$
• $a_{23} = 8+7i$ and $a_{32} = 8-7i$
• The diagonal elements are real.

A fundamental property of Hermitian matrices is that all their eigenvalues (and hence the determinant, which is the product of the eigenvalues) are real. Thus, the determinant $z$ is purely real.