Question

Find the determinate of

(1 1 1
X Y Z
X2 Y2 Z2)

Answer 1

[AI-generated answer — Bot GPT 3.5]

The determinate of a 3x3 matrix can be found by expanding along the first row, first column, or any other row or column.

Expanding along the first row:
det = 1(YZ2 - Y2Z) - 1(XZ2 - X2Z) + 1(XY2 - X2Y)
det = YZ2 - Y2Z - XZ2 + X2Z + XY2 - X2Y

Expanding along the first column:
det = 1(YZ2 - Y2Z) - X(Z2 - 2Z) + X2(Y - Y2)
det = YZ2 - Y2Z - XZ2 + X2Z + XY - XY2

The determinate is the same regardless of how you expand it, and thus the result is:
det = YZ2 - Y2Z - XZ2 + X2Z + XY2 - X2Y