`|(ax,by,cz),(x^2,y^2,z^2),(1,1,1)|=|(a,b,c),(x,y,z),(yz,zx,xy)|`. True or False

Answer

To determine whether the statement \( |(ax, by, cz), (x^2, y^2, z^2), (1, 1, 1)| = |(a, b, c), (x, y, z), (yz, zx, xy)| \) is true or false, we will calculate the determinants of both matrices and compare them.

### Step 1: Calculate the determinant of the first matrix

The first matrix is:
\[ \begin{vmatrix} ax & by & cz \\ x^2 & y^2 & z^2 \\ 1 & 1 & 1 \end{vmatrix} \]

We will expand this determinant using the first row:
\[ = ax \begin{vmatrix} y^2 & z^2 \\ 1 & 1 \end{vmatrix} - by \begin{vmatrix} x^2 & z^2 \\ 1 & 1 \end{vmatrix} + cz \begin{vmatrix} x^2 & y^2 \\ 1 & 1 \end{vmatrix} \]

Calculating the 2x2 determinants:
\[ \begin{vmatrix} y^2 & z^2 \\ 1 & 1 \end{vmatrix} = y^2 - z^2 \]
\[ \begin{vmatrix} x^2 & z^2 \\ 1 & 1 \end{vmatrix} = x^2 - z^2 \]
\[ \begin{vmatrix} x^2 & y^2 \\ 1 & 1 \end{vmatrix} = x^2 - y^2 \]

Substituting these back into the determinant:
\[ = ax(y^2 - z^2) - by(x^2 - z^2) + cz(x^2 - y^2) \]

### Step 2: Simplify the expression

Expanding the expression:
\[ = axy^2 - axz^2 - bxy^2 + byz^2 + czx^2 - czy^2 \]
Combining like terms:
\[ = (ax - by)y^2 + (by - cz)z^2 + czx^2 \]

### Step 3: Calculate the determinant of the second matrix

The second matrix is:
\[ \begin{vmatrix} a & b & c \\ x & y & z \\ yz & zx & xy \end{vmatrix} \]

We will expand this determinant using the first row:
\[ = a \begin{vmatrix} y & z \\ zx & xy \end{vmatrix} - b \begin{vmatrix} x & z \\ yz & xy \end{vmatrix} + c \begin{vmatrix} x & y \\ yz & zx \end{vmatrix} \]

Calculating the 2x2 determinants:
\[ \begin{vmatrix} y & z \\ zx & xy \end{vmatrix} = y \cdot xy - z \cdot zx = xy^2 - z^2x \]
\[ \begin{vmatrix} x & z \\ yz & xy \end{vmatrix} = x \cdot xy - z \cdot yz = x^2y - z^2y \]
\[ \begin{vmatrix} x & y \\ yz & zx \end{vmatrix} = x \cdot zx - y \cdot yz = x^2z - y^2z \]

Substituting these back into the determinant:
\[ = a(xy^2 - z^2x) - b(x^2y - z^2y) + c(x^2z - y^2z) \]

### Step 4: Simplify the expression

Expanding the expression:
\[ = axy^2 - az^2x - bx^2y + bz^2y + cx^2z - cy^2z \]
Combining like terms:
\[ = (a - b)x^2y + (b - c)z^2y + czx^2 \]

### Step 5: Compare the two determinants

Now we compare the two simplified determinants:
1. From the first matrix: \( (ax - by)y^2 + (by - cz)z^2 + czx^2 \)
2. From the second matrix: \( (a - b)x^2y + (b - c)z^2y + czx^2 \)

By inspection, we can see that the terms do not match, indicating that the two determinants are not equal.

### Conclusion

Thus, the statement \( |(ax, by, cz), (x^2, y^2, z^2), (1, 1, 1)| = |(a, b, c), (x, y, z), (yz, zx, xy)| \) is **False**.