Cross product and determinant areas of a Parallelogram

Question

If $x$ and $y$ are two column matrices that make up $A$, a 3x2 matrix, prove that $||x \times y||$ equals $\sqrt{\det(A^TA)}$? How does one go about this proof?

Answer 1

Start by writing $A^TA$ (simple matrix multiplication):$$A^TA=\begin{pmatrix}x^2&x\cdot y\\x\cdot y&y^2\end{pmatrix}$$ You can use Lagrange's identity, written in 3D: $$|x\times y|^2=|x|^2|y|^2-(x\cdot y)^2$$ Alternatively, write explicitly the cross product components, square them, and add together.