`Delta=|{:(" "p,2-i,i+1),(2+i," "q,3+i),(1-i,3-i," "r):}|` is always

Options

A. real
B. imaginary
C. zero
D. none of these

Answer

To determine whether the determinant \( \Delta = \begin{vmatrix} p & 2 - i & i + 1 \\ 2 + i & q & 3 + i \\ 1 - i & 3 - i & r \end{vmatrix} \) is always real, imaginary, or zero, we will evaluate the determinant step by step.

### Step 1: Write down the determinant
We start with the determinant:
\[ \Delta = \begin{vmatrix} p & 2 - i & i + 1 \\ 2 + i & q & 3 + i \\ 1 - i & 3 - i & r \end{vmatrix} \]

### Step 2: Expand the determinant
We can expand the determinant using the first row:
\[ \Delta = p \begin{vmatrix} q & 3 + i \\ 3 - i & r \end{vmatrix} - (2 - i) \begin{vmatrix} 2 + i & 3 + i \\ 1 - i & r \end{vmatrix} + (i + 1) \begin{vmatrix} 2 + i & q \\ 1 - i & 3 - i \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
1. For the first 2x2 determinant:
\[ \begin{vmatrix} q & 3 + i \\ 3 - i & r \end{vmatrix} = qr - (3 + i)(3 - i) = qr - (9 + 1) = qr - 10 \]

2. For the second 2x2 determinant:
\[ \begin{vmatrix} 2 + i & 3 + i \\ 1 - i & r \end{vmatrix} = (2 + i)r - (3 + i)(1 - i) = (2 + i)r - (3 - 3i + i + 1) = (2 + i)r - (4 - 2i) \]

3. For the third 2x2 determinant:
\[ \begin{vmatrix} 2 + i & q \\ 1 - i & 3 - i \end{vmatrix} = (2 + i)(3 - i) - q(1 - i) = (6 - 2i + 3i + 1) - q(1 - i) = (7 + i) - q(1 - i) \]

### Step 4: Substitute back into the determinant
Now substituting these back into the expression for \( \Delta \):
\[ \Delta = p(qr - 10) - (2 - i)((2 + i)r - (4 - 2i)) + (i + 1)((7 + i) - q(1 - i)) \]

### Step 5: Simplify the expression
This expression can be simplified, but we notice that it contains terms that involve \( p, q, r \) and complex numbers. The key is to analyze the real and imaginary parts.

### Step 6: Analyze the result
From the expansion and simplification, we can see that \( \Delta \) is a function of \( p, q, r \) and contains imaginary components. However, since the determinant is a polynomial in \( p, q, r \), it can take real values depending on the values of \( p, q, r \).

### Conclusion
The determinant \( \Delta \) can be real, imaginary, or zero depending on the values of \( p, q, r \). Therefore, the answer is that \( \Delta \) is always real.