Evaluate `|(3+2i,-6i),(2i,3-2i)|,i=sqrt(-1)`

Answer

To evaluate the determinant of the matrix \( |(3 + 2i, -6i), (2i, 3 - 2i)| \), we will follow these steps:

### Step 1: Write down the determinant formula
The determinant of a 2x2 matrix \( |(a, b), (c, d)| \) is calculated using the formula:
\[ \text{det} = ad - bc \]
In our case, \( a = 3 + 2i \), \( b = -6i \), \( c = 2i \), and \( d = 3 - 2i \).

### Step 2: Substitute the values into the formula
Now we substitute the values into the determinant formula:
\[ \text{det} = (3 + 2i)(3 - 2i) - (-6i)(2i) \]

### Step 3: Calculate \( (3 + 2i)(3 - 2i) \)
Using the difference of squares:
\[ (3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - 4(-1) = 9 + 4 = 13 \]

### Step 4: Calculate \( (-6i)(2i) \)
Now calculate the second part:
\[ (-6i)(2i) = -12i^2 = -12(-1) = 12 \]

### Step 5: Combine the results
Now substitute back into the determinant:
\[ \text{det} = 13 - 12 = 1 \]

### Final Answer
Thus, the value of the determinant is:
\[ \boxed{1} \]