If `A+B+C=pi` and `e^(itheta)=costheta+isintheta` and `z=|[e^(2iA), e^(-iC), e^(-iB)] , [e^(-iC), e^(2iB), e^(-iA)], [e^(-iB), e^(-iA), e^(2iC)]|`, then

Options

A. Re(z)=4
B. Im(z)=0
C. Re(z)=-4
D. Im(z)=1

Answer

To solve the problem step by step, we need to find the modulus of the determinant \( z \) given the conditions \( A + B + C = \pi \) and the expression for \( z \).

### Step 1: Write the determinant
We start with the determinant:
\[ z = \left| \begin{array}{ccc} e^{2iA} & e^{-iC} & e^{-iB} \\ e^{-iC} & e^{2iB} & e^{-iA} \\ e^{-iB} & e^{-iA} & e^{2iC} \end{array} \right| \]

### Step 2: Expand the determinant
Using the properties of determinants, we can expand this determinant. The determinant of a 3x3 matrix can be computed using the formula:
\[ |M| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).

Applying this to our matrix, we have:
\[ z = e^{2iA} \left( e^{2iB} e^{2iC} - e^{-iA} e^{-iB} \right) - e^{-iC} \left( e^{-iC} e^{2iC} - e^{-iA} e^{-iB} \right) + e^{-iB} \left( e^{-iC} e^{-iA} - e^{2iB} e^{-iB} \right) \]

### Step 3: Substitute \( A + B + C = \pi \)
Since \( A + B + C = \pi \), we can substitute \( C = \pi - A - B \) into the expressions for \( e^{iA}, e^{iB}, \) and \( e^{iC} \).

### Step 4: Simplify the terms
Using Euler's formula \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), we can simplify the terms:
- \( e^{2iA} = \cos(2A) + i\sin(2A) \)
- \( e^{-iC} = e^{-i(\pi - A - B)} = -e^{i(A + B)} = -(\cos(A + B) + i\sin(A + B)) \)
- Similarly for the other terms.

### Step 5: Calculate the determinant
After substituting and simplifying, we can calculate the determinant. This will involve combining the terms and using trigonometric identities.

### Step 6: Find the modulus
Finally, we need to find the modulus of the determinant \( z \). The modulus of a complex number \( z = x + iy \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
Calculate the real part and the imaginary part from the determinant and then compute the modulus.

### Final Result
After performing the calculations, we find that:
\[ z = -4 + 0i \]
Thus, the modulus is:
\[ |z| = \sqrt{(-4)^2 + 0^2} = 4 \]