Where does this proof that the abs. value of the determinant of a 2x2 matrix A is the area of the image of the unit square under A go wrong? If we have a matrix $A\in M_2(\mathbb{R})$, $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $det(A)=ad-bc$.   $A$ sends the unit square to the parallelogram whose sides are the vectors $(a, c)$ and $(c, d)$. Since the area of a parallelogram is the product of its two side lengths, the area of this parallelogram is $\|(a, c)\|\|(b,d)\|=\sqrt{a^2+c^2}\sqrt{b^2+d^2}=\sqrt{(ab)^2+(ad)^2+(cb)^2+(cd)^2}$. If this is the absolute value of $det(A)$, then it should equal $\mid ad-bc \mid$. But I just don't see how it does . . .  

The area of a parallelogram isn't the product of the two side lengths, it's the product of its base and height. If $\theta$ is the angle between adjacient sides, then the area becomes $||(a,c)||\, ||(b,d)||\sin(\theta)$. 
