Interpreting determinants in terms of area | Matrices | Precalculus | Khan Academy

So, I have a two by two matrix here, and we could view it as having two column vectors. The first column can define this vector (3, 1), which I've depicted in blue here. Then, that second column you can view it as telling us that we have another vector (1, 2), which I have depicted in this pink color.

Now, an interesting interpretation of the determinant of this 2 by 2 matrix is that the absolute value of the determinant is the area of the parallelogram defined by these two vectors. What do I mean by the parallelogram defined by these two vectors? Well, imagine taking this bottom vector and shifting it so its tail is at the head of this pink vector. So, it would look like this — hand-draw it, so it would look something like that.

Then, if you were to take this pink vector and copy it, and shift it up and to the right, so its tail is at the head of the original blue vector, it's going to look something like that. You can see you can use that technique to take any two vectors in the coordinate plane, and they will define a parallelogram. It turns out that the area of this parallelogram is going to be equal to the absolute value of the determinant of this matrix here.

So, what's that going to be? Well, we know how to figure out the determinant: it is 3 times 2, which is 6, minus 1 times 1, which is 1, which is equal to 5. Of course, the absolute value of 5 is 5. Now, that's pretty cool in and of itself. We've figured out one interpretation of a determinant, which will be useful as we build up our understanding of matrices.

But another interpretation is to say, all right, what if A is a transformation matrix? I'm just rewriting it, so we know what a transformation matrix does; it tells us what to do with the unit vectors, so to speak. For example, I have this vector right over here, which is the vector (1, 0). We know that a transformation matrix says all right, take that (1, 0) vector and turn it into the (3, 1) vector; so turn that into that right over there.

We know we have the other— or another unit vector. Let's call this right over here the (0, 1) vector, which goes zero in the x direction and one in the y direction. The transformation matrix says, hey, turn that into the (1, 2) vector. But then, you could think about it—it's also not just transforming the individual vectors; it's also scaling up the area defined by the vectors.

So, the area defined by these two original—we could say unit vectors— we can see it's one by one; it's that area right over there. So, this transformation is taking us from an area of one to an area of five. It's scaling it up by a factor of five. Now, that's reasonably interesting just for this one unit square.

But because of that, it'll actually scale up the area of any figure. Let's say you had a figure like this— this type of oval circle thing; it has some area. If you apply this transformation matrix, it will look something like this—I'm just approximating it. It would look something like that.

So, this will tell us that this bigger blob has five times the area of this original blue blob, because the bigger blob is the image once we've transformed the smaller blob by this transformation matrix. So, if I were to tell you that the area of this smaller circle is, let's say, 0.6, but then we were to apply the transformation, and someone were to ask you what is the area of this bigger blob, which is the image of the smaller circle after the transformation, well, you would take 0.6 multiplied by the absolute value of the determinant of the transformation matrix.

You'd multiply it by 5. So, 0.6 times 5 would be 3 square units. A hint at the reason why this works is that any region on the coordinate plane can be represented as a series of squares, and then if you apply the transformation, you're really just transforming each of those individual squares. So, the scaling up of the area would be the same scale you do to any one of those smaller squares.