`" if " |{:(yz-x^(2),,zx-y^(2),,xy-z^(2)),(xz-y^(2),,xy-z^(2),,yz-x^(2)),(xy-z^(2),,yz-x^(2),,zx-y^(2)):}|=|{:(r^(2),,u^(2),,u^(2)),(u^(2),,r^(2),,u^(2)),(u^(2),,u^(2),,r^(2)):}|` then

Options

A. `r^(2)=x+y+z`
B. `r^(2) =x^(2) =y^(2) +z^(2)`
C. `u^(2) =yz+zx+xy`
D. `u^(2) =xyz`

Explanation

To solve the given determinant equation, we start with the two determinants provided:

1. Left-hand side determinant:
\[ D_1 = \begin{vmatrix} yz - x^2 & zx - y^2 & xy - z^2 \\ xz - y^2 & xy - z^2 & yz - x^2 \\ xy - z^2 & yz - x^2 & zx - y^2 \end{vmatrix} \]

2. Right-hand side determinant:
\[ D_2 = \begin{vmatrix} r^2 & u^2 & u^2 \\ u^2 & r^2 & u^2 \\ u^2 & u^2 & r^2 \end{vmatrix} \]

We need to establish a relationship between \( r, u \) and \( x, y, z \) based on the equality of these determinants.

### Step 1: Calculate the determinant \( D_2 \)

The determinant \( D_2 \) can be calculated using the formula for the determinant of a \( 3 \times 3 \) matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix:
\[ D_2 = r^2 \begin{vmatrix} r^2 & u^2 \\ u^2 & r^2 \end{vmatrix} - u^2 \begin{vmatrix} u^2 & u^2 \\ u^2 & r^2 \end{vmatrix} + u^2 \begin{vmatrix} u^2 & r^2 \\ u^2 & u^2 \end{vmatrix} \]

Calculating these \( 2 \times 2 \) determinants:
1. \( \begin{vmatrix} r^2 & u^2 \\ u^2 & r^2 \end{vmatrix} = r^4 - u^4 \)
2. \( \begin{vmatrix} u^2 & u^2 \\ u^2 & r^2 \end{vmatrix} = u^2r^2 - u^4 = u^2(r^2 - u^2) \)
3. \( \begin{vmatrix} u^2 & r^2 \\ u^2 & u^2 \end{vmatrix} = u^4 - u^2r^2 = u^2(u^2 - r^2) \)

Substituting these back into \( D_2 \):
\[ D_2 = r^2(r^4 - u^4) - u^2[u^2(r^2 - u^2)] + u^2[u^2(u^2 - r^2)] \]
This simplifies to:
\[ D_2 = r^6 - r^2u^4 - u^4 + u^4 = r^6 - r^2u^4 \]

### Step 2: Analyze the left-hand side determinant \( D_1 \)

The left-hand side determinant \( D_1 \) is more complex. However, we can observe that it is structured similarly to the cofactor matrix of \( x, y, z \).

### Step 3: Establish the relationship

From the video transcript, we find that:
\[ D_1 = (x^2 + y^2 + z^2)^2 - (xy + yz + zx)^2 \]
This can be expressed as:
\[ D_1 = (x^2 + y^2 + z^2 - (xy + yz + zx))(x^2 + y^2 + z^2 + (xy + yz + zx)) \]

### Step 4: Set the determinants equal

Since \( D_1 = D_2 \), we have:
\[ (x^2 + y^2 + z^2 - (xy + yz + zx))(x^2 + y^2 + z^2 + (xy + yz + zx)) = r^6 - r^2u^4 \]

### Step 5: Compare coefficients

By comparing coefficients from both sides, we can derive relationships:
- \( x^2 + y^2 + z^2 = r^2 \)
- \( xy + yz + zx = u^2 \)

### Conclusion

Thus, the relationships derived are:
\[ x^2 + y^2 + z^2 = r^2 \quad \text{and} \quad xy + yz + zx = u^2 \]