A simple introduction to a determinant is that it’s the area of a box.
Working in two dimensions, I’ll outline
- the geometric picture of a linear transformation
- the geometric picture of a determinant as an area
- how the geometric picture leads to a few important properties of determinants
- how the geometric picture of linear transformations can be expressed with matrices
- how the geometric picture leads us to a formula for the determinant of a matrix
This post is long already. To keep it from becoming even longer, in some places I have had to leave out certain steps in the logic.
We’ll start with the coordinate plane. It’s a grid of points.
Then we scissor it or blow it up or shrink it down. Here are some examples:
To make them, I took the original image and applied the “shear”, “rotate”, and “scale” tools in GIMP (an open-source PhotoShop equivalent). You can try it yourself on any image just by using those tools.
These are called “linear transformations”. To simplify the way we picture them, we can just look at what they do to a box at the origin. I’ll make the box 3×3 so it’s visible, but imagine that each line represents a distance 1/3, so the sides of the box are length 1.
If we wanted, we could use something more complicated:
But since the apple is made from little boxes and all little boxes get treated the same way, we might as well focus on what happens to just one box.
Under any linear transformation, the box turns into a parallelogram.
The area of that parallelogram is called the determinant of the linear transformation.
There’s one extra rule. If the red and blue sides switch (as they would if I used the “flip” tool in GIMP), the determinant is negative. Here’s an example:
Since any area is made from little boxes and each little box’s area gets multiplied by the determinant, the area of any shape at all gets multiplied by the determinant. So for the apple, the determinant is the area of the apple on the right divided by the area of the apple on the left.
So that’s what a determinant is. What remains is to show what it’s about and what it has to do with the matrices you were wondering about.
Let’s look at some properties first. Imagine doing two transformations in a row. We’ll call this “multiplying the transformations”. The result is just another linear transformation.
When we do these sequential transformations, the area of our box gets multiplied by the determinant each time. If the determinant of the first transformation is 3 and the determinant of the second transformation is 5, the area gets multiplied by 15 overall, so the determinant of the combined transformation is 15. Multiplying transformations means multiplying determinants.
Next we’ll think about inverses. An inverse is a transformation that takes you back to where you started. The inverse of a transformation that rotates 45 degrees clockwise and multiplies everything by 2 is a transformation that rotates 45 degrees counterclockwise and cuts everything in half.
The determinant of the first transformation is 4 because each side of the box is doubled. The determinant of the second transformation is 1/4.
This is a general rule. Suppose two transformations are inverses. Then their determinants must multiply to 1, because the area of the box doesn’t change overall.
Next suppose a transformation’s determinant is zero. Then it doesn’t have an inverse because any number times zero is still zero, so there’s no transformation that takes the determinant back to one.
Geometrically, a transformation with zero determinant collapses everything to a line.
The line doesn’t have to be flat like this. It could be at any angle. Also, I didn’t collapse this completely to a line, since then you couldn’t see it. Transformations with zero determinant are bad news.
- Linear transformations are some combination of the “scale”, “rotate”, “shear”, and “flip” tools in Photoshop.
- The determinant of a linear transformation is the factor by which the transformation changes the area.
- The determinants of inverse transformations multiply to 1.
- If the determinant is zero, the matrix doesn’t have an inverse. (The converse of this also holds, although we didn’t discuss it.)
Let’s move on to matrices. Take a linear transformation like this:
If we superimpose the original onto the final, we can see the coordinates of the new parallelogram in terms of the original grid.
We can describe the transformation completely using four numbers, two for the coordinates of the blue side and two for the coordinates of the red side. We’ll call those numbers .
We’ll represent points with column matrices. So the point will be represented by the matrix . (A matrix doesn’t have to be square. This is a 2×1 matrix.)
With this notation, we can represent our linear transformation by
This actually represents the entire transformation, even though it looks like we’ve only looked at two points. The reason is that any other point is made up out of the two we’ve already examined. For example
There’s a much more convenient way to write all this, which is in the form of a 2×2 matrix. , which is the blue part of our parallelogram, becomes the first column of the matrix. is the second column.
We can view matrix multiplication as
Check that this works for the example of .
You may have learned to do this multiplication one row at a time rather than one column at a time. The result is the same.
This shows how a matrices describe linear transformations. All that remains is to tie in the concept of a determinant.
Remembering that a determinant is the area of a box, we can find a formula for the determinant by looking at some properties of area.
The area of the original 1×1 box is 1. That means
because that’s the identity matrix. It’s the linear transformation that does nothing. (The vertical lines around the matrix indicate that we’re taking a determinant.)
When we switch the blue and red sides of the box, the determinant is -1. The matrix that does this is
When we multiply the blue side by two, the determinant gets multiplied by that same factor. Since this is represented in the matrix by multiplying the first column by two, we have
This matrix is not invertible. It collapse everything onto the x-axis, making a “box” of zero area, so its determinant is zero. Similarly,
The final property we need of determinants/areas is linearity. Check out this picture:
It requires a little explanation. There are three linear transformations here, all sharing the same red side. The first two have the blue and purple sides. These are smaller. When we add them up, we get the third one with the gray side, so this picture represents adding linear transformations (which is different than multiplying them.) The green area is the area of the big transformation with the gray side.
The two smaller ones, with the blue and purple sides, have a total area equal to the green area. We can see this because there is a triangle of stuff that’s outside the green area, and therefore not counted. However, there’s also a triangle of extra stuff in the green area that’s not part two smaller parallelograms. These two triangles have the same area and cancel each other out, so that the small parallelograms have the same total area as the single big one.
Translating this into matrices means we can add determinants when one column is shared. This is called linearity in a column. For example
So the properties we found are
- The determinant of the identity is one.
- The determinant of the matrix that switches horizontal and vertical is -1.
- Multiplying a column by a number multiplies the determinant by that number.
- The determinant is linear in a column.
These properties combined let us find the determinant of any matrix. Start with
use linearity in the first column to write this as
now use linearity in the second column to make it
We have already set up the tools to evaluate each of these individually. The determinant is
That’s the area of the parallelogram. You could find it by other geometrical means, too, but knowing the formula for the determinant makes it easy.