If `1 + (1)/(a) + (1)/(b) + (1)/(c) = 0`, then
`Delta = |(1 +a,1,1),(1,1 +b,1),(1,1,1 +c)|` is equal to

Options

A. 0
B. abc
C. `-abc`
D. none of these

Answer

To solve the given problem, we need to evaluate the determinant \( \Delta = |(1 + a, 1, 1), (1, 1 + b, 1), (1, 1, 1 + c)| \) under the condition \( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0 \).

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

\[ \Delta = \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} \]

### Step 2: Apply column operations
We can simplify the determinant by performing column operations. Let's subtract the second column from the first column:

\[ C_1 \rightarrow C_1 - C_2 \]

This gives us:

\[ \Delta = \begin{vmatrix} (1 + a - 1) & 1 & 1 \\ (1 - (1 + b)) & 1 + b & 1 \\ (1 - 1) & 1 & (1 + c - 1) \end{vmatrix} \]

This simplifies to:

\[ \Delta = \begin{vmatrix} a & 1 & 1 \\ -b & 1 + b & 1 \\ 0 & 1 & c \end{vmatrix} \]

### Step 3: Apply another column operation
Now, let's perform another column operation by subtracting the third column from the second column:

\[ C_2 \rightarrow C_2 - C_3 \]

This gives us:

\[ \Delta = \begin{vmatrix} a & 0 & 1 \\ -b & b & 1 \\ 0 & 0 & c \end{vmatrix} \]

### Step 4: Expand the determinant
Now we can expand the determinant along the first column:

\[ \Delta = a \begin{vmatrix} b & 1 \\ 0 & c \end{vmatrix} - (-b) \begin{vmatrix} 0 & 1 \\ 0 & c \end{vmatrix} + 0 \]

Calculating the 2x2 determinants, we have:

\[ \Delta = a(b \cdot c - 0) + 0 = abc \]

### Step 5: Substitute the given condition
Now, we need to use the condition given in the problem:

From the condition \( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0 \), we can rearrange it to:

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = -1 \]

This implies:

\[ \frac{bc + ac + ab}{abc} = -1 \]

Thus, we have:

\[ bc + ac + ab = -abc \]

### Step 6: Final result
Substituting this back into our determinant expression, we find:

\[ \Delta = abc \cdot 0 = 0 \]

Therefore, the value of the determinant \( \Delta \) is:

\[ \Delta = 0 \]