yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Finding area of figure after transformation using determinant | Matrices | Khan Academy


2m read
·Nov 10, 2024

We're told to consider this matrix transformation. This is a matrix that you can use, it represents a transformation on the entire coordinate plane. Then they tell us that the transformation is performed on the following rectangle. So, this is the rectangle before the transformation. They say, what is the area of the image of the rectangle under this transformation? The image of the rectangle is what the rectangle becomes after the transformation.

So pause this video and see if you can answer that before we work through it on our own. All right, so the main thing to realize is if we have a matrix transformation or a transformation matrix like this, if we take the absolute value of its determinant, that value tells us how much that transformation scales up areas of figures.

So let's just do that. Let's evaluate the absolute value of the determinant here. The absolute value of the determinant would be the absolute value of five times eight minus nine times four. Remember, for a two by two matrix, the determinant is just this times this minus this times that. That's going to be the absolute value of 40 minus 36, which is just the absolute value of 4, which is just going to be equal to 4.

So, this tells us that this transformation will scale up area by a factor of 4. So what's the area before the transformation? Well, we can see that this is, let's see, it's 5 units tall and it is 7 units wide. So this has an area of 35 square units pre-transformation.

So post-transformation, we just multiply it by the absolute value of the determinant to get, let's see, 4 times 35, which is 140 square units. And we're done.

More Articles

View All
Measuring area with partial unit squares | Math | 3rd grade | Khan Academy
Each square in the grid is a unit square with an area of 1 square cm. So, each of these squares is 1 square cm. This is 1 square cm, and this is 1 square cm, and so on. Now we’re asked, what is the area of the figure? By figure, I’m sure they mean this bl…
Ice Age Cave Art: Unlocking the Mysteries Behind These Markings | Nat Geo Live
Genevieve Von Petzinger: This incredible art that mostly dates between 10 and 40,000 years ago. What we often think of, of course, is the animals. But there’s this other enormous group called the geometric signs that outnumber the animals and the humans a…
Median, mean and skew from density curves | AP Statistics | Khan Academy
In other videos, we introduce ourselves to the idea of a density curve, which is a summary of a distribution—a distribution of data. In the future, we’ll also look at things like probability density. But what I want to talk about in this video is to thin…
How Many Countries Are There?
How many countries are there? Easy: just grab a map and start counting, yes? No. Not all maps are created equal – borders will differ depending on who you got the map from. So if individuals disagree, then surely a committee will save the day. Go to the U…
Rebuilding the World of 1620 | Saints & Strangers
I’ve covered myself a little. I do not sleep safe, nor do I seek glory at war. If it’s something like this, where it’s 1620, you finally got to get yourself immersed into the era. To start with, I did a lot of research on the pilgrims themselves: who they…
Examples of bias in surveys | Study design | AP Statistics | Khan Academy
We’re told that David hosts a podcast, and he’s curious how much his listeners like his show. He decides to start with an online poll. He asks his listeners to visit his website and participate in the poll. The poll shows that 89 percent of about 200 resp…