`|(1+i, 1-i, i), (1-i, i, 1+i), (i, 1+i, 1-i)|=`

Options

A. `-4-7i`
B. `4+7i`
C. `3+7i`
D. `7+4i`

Answer

To find the determinant of the matrix

\[ \begin{pmatrix} 1+i & 1-i & i \\ 1-i & i & 1+i \\ i & 1+i & 1-i \end{pmatrix} \]

we will use the formula for the determinant of a 3x3 matrix. The determinant \(D\) of a matrix

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

is given by:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

### Step 1: Identify the elements of the matrix

From our matrix, we have:

- \(a = 1+i\)
- \(b = 1-i\)
- \(c = i\)
- \(d = 1-i\)
- \(e = i\)
- \(f = 1+i\)
- \(g = i\)
- \(h = 1+i\)
- \(i = 1-i\)

### Step 2: Calculate the determinant using the formula

Using the determinant formula:

\[ D = (1+i) \left( i(1-i) - (1+i)(1+i) \right) - (1-i) \left( (1-i)(1-i) - i(1+i) \right) + i \left( (1-i)(1+i) - i(i) \right) \]

### Step 3: Calculate each term in the determinant

1. **First term**:
\[ (1+i) \left( i(1-i) - (1+i)(1+i) \right) \]
- Calculate \(i(1-i) = i - i^2 = i + 1\)
- Calculate \((1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i\)
- Thus, \(i(1-i) - (1+i)(1+i) = (i + 1) - 2i = 1 - i\)
- Therefore, the first term becomes \((1+i)(1-i) = 1^2 - i^2 = 1 + 1 = 2\)

2. **Second term**:
\[ -(1-i) \left( (1-i)(1-i) - i(1+i) \right) \]
- Calculate \((1-i)(1-i) = 1 - 2i + i^2 = 1 - 2i - 1 = -2i\)
- Calculate \(i(1+i) = i + i^2 = i - 1\)
- Thus, \(-2i - (i - 1) = -2i - i + 1 = 1 - 3i\)
- Therefore, the second term becomes \(-(1-i)(1-3i) = -(1 - 3i + i - 3i^2) = -(1 - 3i + i + 3) = -4 + 2i\)

3. **Third term**:
\[ i \left( (1-i)(1+i) - i(i) \right) \]
- Calculate \((1-i)(1+i) = 1 - i^2 = 1 + 1 = 2\)
- Calculate \(i(i) = i^2 = -1\)
- Thus, \(2 - (-1) = 2 + 1 = 3\)
- Therefore, the third term becomes \(i \cdot 3 = 3i\)

### Step 4: Combine all terms

Now we combine all the terms:

\[ D = 2 + (-4 + 2i) + 3i = 2 - 4 + 2i + 3i = -2 + 5i \]

### Final Result

Thus, the determinant of the given matrix is:

\[ \boxed{-2 + 5i} \]