In his video on determinants, 3blue1brown provides a nice geometric interpretation for the determinant of a matrix,
Generally, the determinant represents the factor by which a matrix scales the area/volume/etc after a linear transformation.
This is a better approach than rote memorization of the formula for calculating a determinant and just rolling with it (like I did for years).
from IPython.display import Image Image('images/determinant_formula_2d.PNG')
To see why this is the case, let’s consider a couple simple shapes/transformations.
As mentioned in our Linear Transformation notebook, we consider the matrix
A as the effect on our basis unit vectors
j. The area of a
1x1 rectangle is obviously
1, so looking at a simple matrix below that stretches our area in a straight line about the X and Y axes, we can see that the form expands to area
a * d, which is the same as the growth of scale when dividing by
Making the transformation a bit spicier, we’ll make our transformation on
j one in two dimensions, keeping
i in one.
The area of a parallelogram is just the height multiplied by the width, which is still easy enough to see in this form.
Finally, making the transformation on both
j be across two dimensions, 3b1b provided a really elegant geometric representation for the formula that you might have memorized to show that this still holds.
There is, of course, a general form for calculating the determinant, but that’s easily obviated away with
Remembering the intuition behind the determinant is far more valueable, imo.
Transformation Non Unit Vector Shapes
As remarked in the notebook on Linear Transformations, the consequence of applying a linear transformation using a matrix
A is that parallel and evenly-spaced points remain so after a transformation. And so the scaling of our area follows the same effect for any shape. e.g.