A linear transformation given by a 2x2 matrix A doubles areas. What is the determinant of A? How should I think about this problem?
 A: Determinants are inherently defined to be the scaling factor,multiplying by which increases/decreases the unit square in Cartesian coordinate by some fraction. Therefore if on multiplication by such an operator doubles the area, you can say without any calculation that it's determinant is plus or minus 2, minus being the case when the directions of the axes are reversed. Similarly if some operator (matrix) increases the area three times, it's determinant is plus or minus 3.
If you wonder why,you should watch 3Blue1brown's video on determinants.
A: Consider the area of the unit square $|[i\times j]|=1$.
If the transformation is $\begin{pmatrix}x\\y\end{pmatrix}\to
\begin{pmatrix}a&b\\c&d\end{pmatrix}
\begin{pmatrix}x\\y\end{pmatrix}$ then it maps $i\to ai+cj$, $j\to bi+dj$ and the area of the image will be $$|[(ai+cj)\times(bi+dj)]|\\
=|[ai\times bi]+[ai\times dj]+[cj\times bi]+[cj\times dj]|\\
=|0+ad[i\times j]+bc[j\times i]+0|\\
=|ad-bc|\cdot|[i\times j]|=|\det A|=2$$
So $\det A=\pm 2$.
Another way to think of it is $A=SJS^{-1}$ where $S$ is some rotation matrix and $J$ is some scale matrix, so if $J$ scales one coordinate in $\lambda_x$ times and another in $\lambda_y$ times, the signed area scale will be $\lambda_x\lambda_y$, but $\det S=1$ thus $\det A=\det J=\lambda_x\lambda_y$ that makes the problem simple.
