If z = `|{:(-5,3+4i,5-7i),(3-4i,6,8+7i),(5+7i,8-7i,9):}|`, then z is

Options

A. purely real
B. purely imaginary
C. a + ib, where a `ne` 0, `b ne 0 `
D. a + ib, where b = 4

Answer

To find the value of \( z \) given the determinant:

\[ z = \begin{vmatrix} -5 & 3 + 4i & 5 - 7i \\ 3 - 4i & 6 & 8 + 7i \\ 5 + 7i & 8 - 7i & 9 \end{vmatrix} \]

we will calculate the determinant step by step.

### Step 1: Write the Determinant
We start by writing the determinant explicitly:

\[ z = \begin{vmatrix} -5 & 3 + 4i & 5 - 7i \\ 3 - 4i & 6 & 8 + 7i \\ 5 + 7i & 8 - 7i & 9 \end{vmatrix} \]

### Step 2: Apply the Determinant Formula
The determinant of a \( 3 \times 3 \) matrix is given by:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

where the matrix is:

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

For our matrix, we have:

- \( a = -5 \), \( b = 3 + 4i \), \( c = 5 - 7i \)
- \( d = 3 - 4i \), \( e = 6 \), \( f = 8 + 7i \)
- \( g = 5 + 7i \), \( h = 8 - 7i \), \( i = 9 \)

### Step 3: Calculate the Determinant
Now we can substitute these values into the determinant formula:

\[ z = -5 \left( 6 \cdot 9 - (8 + 7i)(8 - 7i) \right) - (3 + 4i) \left( (3 - 4i) \cdot 9 - (5 + 7i)(8 - 7i) \right) + (5 - 7i) \left( (3 - 4i)(8 - 7i) - 6(5 + 7i) \right) \]

### Step 4: Calculate Each Term
1. Calculate \( 6 \cdot 9 = 54 \).
2. Calculate \( (8 + 7i)(8 - 7i) = 64 + 49 = 113 \).
3. Thus, \( 6 \cdot 9 - (8 + 7i)(8 - 7i) = 54 - 113 = -59 \).

Now substituting back into the determinant:

\[ z = -5(-59) - (3 + 4i) \left( (3 - 4i) \cdot 9 - (5 + 7i)(8 - 7i) \right) + (5 - 7i) \left( (3 - 4i)(8 - 7i) - 6(5 + 7i) \right) \]

### Step 5: Simplify Further
Continue simplifying the remaining terms.

1. Calculate \( (3 - 4i) \cdot 9 = 27 - 36i \).
2. Calculate \( (5 + 7i)(8 - 7i) = 40 - 35i + 56i - 49 = 40 + 21i - 49 = -9 + 21i \).
3. Thus, \( (3 - 4i) \cdot 9 - (5 + 7i)(8 - 7i) = (27 - 36i) - (-9 + 21i) = 36 - 57i \).

Now substitute back into the determinant:

\[ z = 295 - (3 + 4i)(36 - 57i) + (5 - 7i) \left( (3 - 4i)(8 - 7i) - 6(5 + 7i) \right) \]

### Step 6: Final Calculation
Continue calculating the remaining terms, and after simplification, you will find that \( z \) is a purely real number.

### Conclusion
The final value of \( z \) is a real number, confirming that the determinant of this Hermitian matrix is real.