Without expanding at any stage, find the value of : `|{:(,a,b,c),(,a+2x, b+2y, c+2z),(,x,y,z):}|`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z \end{vmatrix} \]

we will perform row operations without expanding the determinant.

### Step 1: Write down the determinant

We start with the determinant as given:

\[ D = \begin{vmatrix} a & b & c \\ a + 2x & b + 2y & c + 2z \\ x & y & z \end{vmatrix} \]

### Step 2: Perform row operation on \( R_2 \)

We will modify the second row \( R_2 \) by subtracting \( 2 \times R_3 \) from it. This operation can be expressed as:

\[ R_2 \rightarrow R_2 - 2R_3 \]

Calculating this gives:

\[ R_2 = (a + 2x - 2x, b + 2y - 2y, c + 2z - 2z) = (a, b, c) \]

So the determinant now looks like:

\[ D = \begin{vmatrix} a & b & c \\ a & b & c \\ x & y & z \end{vmatrix} \]

### Step 3: Identify identical rows

Now we observe that the first row \( R_1 \) and the second row \( R_2 \) are identical:

\[ R_1 = R_2 = (a, b, c) \]

### Step 4: Conclude the value of the determinant

Since two rows of a determinant are identical, the value of the determinant is zero:

\[ D = 0 \]

Thus, the final answer is:

\[ \boxed{0} \]