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

Options

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

Answer

To solve the determinant equation given by

\[ \begin{vmatrix} y & x & y + z \\ z & y & x + y \\ x & z & z + x \end{vmatrix} = 0, \]

we will follow these steps:

### Step 1: Write the determinant
We start with the determinant:

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

### Step 2: Apply row operations
We can simplify the determinant by performing row operations. Let's add the first row to the second and third rows:

\[ R_2 \rightarrow R_2 + R_1 \quad \text{and} \quad R_3 \rightarrow R_3 + R_1 \]

This gives us:

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

This simplifies to:

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

### Step 3: Factor out common terms
Now, we can factor out \(x + y + z\) from the first column:

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

### Step 4: Evaluate the determinant
We can evaluate the determinant of the remaining matrix. However, we can also set the determinant equal to zero:

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

This implies either \(x + y + z = 0\) or the other expression equals zero.

### Step 5: Solve the equations
1. From \(x + y + z = 0\), we can derive:
\[ x + y = -z \]

2. For the other expression, we can simplify it further (as indicated in the video) and find that it leads to:
\[ z = x \]

### Conclusion
Thus, we have two conditions:
- \(x + y = -z\)
- \(z = x\)

### Step 6: Identify the correct option
Looking at the options provided:
- Option B states: "Either \(x + y = -z\) or \(x = z\)", which matches our derived conditions.

Thus, the correct answer is:

**Option B: Either \(x + y = -z\) or \(x = z\)**.