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

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.