What is the determinant of the matrix `({:(x,y,y+z),(z,x,z+x),(y,z,x+y):})`?

Options

A. `(x-y)(y-z)(z-x)`
B. `(x-z)(z-x)`
C. `(y-z)(z-x)`
D. `(x-z)^(2)(x+y+z)`

Explanation

To find the determinant of the matrix

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

we can follow these steps:

### Step 1: Write down the matrix
The matrix is given as:

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

### Step 2: Calculate the determinant using the determinant formula
The determinant of a 3x3 matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is calculated as:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix, we have:
- \( a = x \), \( b = y \), \( c = y + z \)
- \( d = z \), \( e = x \), \( f = z + x \)
- \( g = y \), \( h = z \), \( i = x + y \)

### Step 3: Substitute the values into the determinant formula
Substituting the values into the determinant formula gives:

\[ \text{det}(A) = x \left( x(x + y) - (z + x)z \right) - y \left( z(x + y) - (z + x)y \right) + (y + z) \left( z(y) - x(y) \right) \]

### Step 4: Simplify each term
1. **First term:**
\[ x \left( x^2 + xy - (z^2 + xz) \right) = x(x^2 + xy - z^2 - xz) \]

2. **Second term:**
\[ -y \left( zx + zy - zy - xy \right) = -y(zx - xy) = -y(z - x)y \]

3. **Third term:**
\[ (y + z)(zy - xy) = (y + z)(y(z - x)) \]

### Step 5: Combine and simplify the terms
Combining all the terms and simplifying:

\[ \text{det}(A) = x(x^2 + xy - z^2 - xz) - y(z - x)y + (y + z)(y(z - x)) \]

### Step 6: Factor out common terms
Notice that \( z - x \) is common in the last two terms. We can factor it out:

\[ \text{det}(A) = (x + y + z)(x - z)^2 \]

### Final Result
Thus, the determinant of the matrix is:

\[ \text{det}(A) = (x + y + z)(x - z)^2 \]