Find the value of : `|{:(1,x,yz),(1,y,zx),(1,z,xy):}|`

Options

A. xyz(x+y+z)
B. (x-y)(y-z)(z-x)
C. xyz(x-y)(y-z)(z-x)
D. none of these

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} 1 & x & y z \\ 1 & y & z x \\ 1 & z & x y \end{vmatrix} \]

we will follow these steps:

### Step 1: Subtract the first row from the second and third rows.

We perform the row operations \( R_2 \rightarrow R_2 - R_1 \) and \( R_3 \rightarrow R_3 - R_1 \):

\[ D = \begin{vmatrix} 1 & x & y z \\ 0 & y - x & z x - y z \\ 0 & z - x & x y - y z \end{vmatrix} \]

### Step 2: Simplify the determinant.

Notice that the first column now has two zeros. We can factor out the common terms from the second and third rows. The determinant simplifies to:

\[ D = 1 \cdot \begin{vmatrix} y - x & z x - y z \\ z - x & x y - y z \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinant.

Now we compute the determinant of the 2x2 matrix:

\[ \begin{vmatrix} y - x & z x - y z \\ z - x & x y - y z \end{vmatrix} = (y - x)(x y - y z) - (z x - y z)(z - x) \]

### Step 4: Expand the determinant.

Expanding this gives:

\[ (y - x)(x y - y z) = y^2 x - y^2 z - x^2 y + x y z \]
\[ (z x - y z)(z - x) = z^2 x - z y z - x y z + x^2 y = z^2 x - y z^2 - x y z + x^2 y \]

### Step 5: Combine the results.

Now we combine the results:

\[ D = y^2 x - y^2 z - x^2 y + x y z - (z^2 x - y z^2 - x y z + x^2 y) \]

### Step 6: Simplify further.

Combining like terms leads us to:

\[ D = y^2 x - y^2 z - z^2 x + y z^2 + x y z - x^2 y + x^2 y - x y z \]

### Step 7: Factor the expression.

The expression can be factored as:

\[ D = (x - y)(y - z)(z - x) \]

Thus, the value of the determinant is:

\[ D = (x - y)(y - z)(z - x) \]

### Final Result:

\[ \boxed{(x - y)(y - z)(z - x)} \]

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