If `|{:(y,x,y+z),(z,y,x+y),(x,z,z+x):}|=0`, then which one of the following is correct ?

Options

A. Either x + y = z or x = y
B. Either x + y = - z or x = z
C. Either x + z = y or z = y
D. Either z + y = x or x = y

Explanation

To solve the problem, we need to analyze the determinant of the given matrix and find the conditions under which it equals zero. The matrix is:

\[ \begin{pmatrix} y & x & y + z \\ z & y & x + y \\ x & z & z + x \end{pmatrix} \]

### Step 1: Set Up the Determinant

We start by writing the determinant of the matrix:

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

### Step 2: Row Operation

We perform the row operation \( R_1 \rightarrow R_1 + R_2 + R_3 \):

\[ R_1 = (y + z + x, x + y + z, (y + z) + (x + y) + (z + x)) \]

This simplifies to:

\[ R_1 = (x + y + z, x + y + z, 2x + 2y + 2z) \]

The new matrix becomes:

\[ \begin{pmatrix} x + y + z & x + y + z & 2(x + y + z) \\ z & y & x + y \\ x & z & z + x \end{pmatrix} \]

### Step 3: Factor Out Common Terms

Now we can factor out \( (x + y + z) \) from the first row:

\[ D = (x + y + z) \begin{vmatrix} 1 & 1 & 2 \\ z & y & x + y \\ x & z & z + x \end{vmatrix} \]

### Step 4: Column Operations

Next, we perform column operations to simplify the determinant further. We can do \( C_2 \rightarrow C_2 - C_1 \) and \( C_3 \rightarrow C_3 - 2C_1 \):

This gives us:

\[ \begin{pmatrix} 1 & 0 & 0 \\ z & y - z & y \\ x & z - x & z + x - 2x \end{pmatrix} \]

### Step 5: Simplifying the Determinant

Now we can compute the determinant of the simplified matrix:

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

### Step 6: Calculate the Determinant

Calculating the determinant using the first row:

\[ D = (x + y + z) \left( 1 \cdot \begin{vmatrix} y - z & y \\ z - x & -x + z \end{vmatrix} \right) \]

Calculating the 2x2 determinant:

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

### Step 7: Setting the Determinant to Zero

Setting the determinant equal to zero gives us:

\[ (x + y + z) \cdot \text{(some expression)} = 0 \]

This implies either:

1. \( x + y + z = 0 \)
2. The other expression must also equal zero.

### Conclusion

From the operations and simplifications, we find that the conditions lead us to conclude that:

- \( x + y + z = 0 \)
- \( z = x \)

Thus, the correct option is:

**Option B: \( z = x \) and \( x + y + z = 0 \)**.