If `|{:(a-b,b-c,c-a),(x-y,y-z,z-x),(p-q,q-r,r-p):}|` is expressible as `lamda|(a,b,c),(x,y,z),(p,q,r)|` then `lamda` is

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To find the value of \( \lambda \) such that

\[ \left| \begin{array}{ccc} a-b & b-c & c-a \\ x-y & y-z & z-x \\ p-q & q-r & r-p \end{array} \right| = \lambda \left| \begin{array}{ccc} a & b & c \\ x & y & z \\ p & q & r \end{array} \right| \]

we will evaluate the left-hand side determinant.

### Step 1: Evaluate the Determinant

We denote the left-hand side determinant as \( D \):

\[ D = \left| \begin{array}{ccc} a-b & b-c & c-a \\ x-y & y-z & z-x \\ p-q & q-r & r-p \end{array} \right| \]

### Step 2: Apply Column Operations

We can simplify this determinant by performing column operations. We will add the second and third columns to the first column:

\[ C_1 \rightarrow C_1 + C_2 + C_3 \]

This gives us:

\[ D = \left| \begin{array}{ccc} (a-b) + (b-c) + (c-a) & b-c & c-a \\ (x-y) + (y-z) + (z-x) & y-z & z-x \\ (p-q) + (q-r) + (r-p) & q-r & r-p \end{array} \right| \]

### Step 3: Simplify the First Column

Now, simplifying the first column:

1. For the first row:
\[ (a-b) + (b-c) + (c-a) = a - b + b - c + c - a = 0 \]

2. For the second row:
\[ (x-y) + (y-z) + (z-x) = x - y + y - z + z - x = 0 \]

3. For the third row:
\[ (p-q) + (q-r) + (r-p) = p - q + q - r + r - p = 0 \]

Thus, the first column becomes:

\[ \left| \begin{array}{ccc} 0 & b-c & c-a \\ 0 & y-z & z-x \\ 0 & q-r & r-p \end{array} \right| \]

### Step 4: Evaluate the Determinant with a Zero Column

Since the first column of the determinant is now all zeros, we can conclude that:

\[ D = 0 \]

### Step 5: Relate to the Right-Hand Side

Now, we have:

\[ 0 = \lambda \left| \begin{array}{ccc} a & b & c \\ x & y & z \\ p & q & r \end{array} \right| \]

This implies that either \( \lambda = 0 \) or the right-hand side determinant is also zero.

### Conclusion

Since we are looking for \( \lambda \) and the determinant on the right-hand side can take any value, the most suitable conclusion is:

\[ \lambda = 0 \]

Thus, the value of \( \lambda \) is:

\[ \boxed{0} \]