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Finding area of figure after transformation using determinant

In this worked example, Sal finds the area of the image of a rectangle after a transformation defined by a given matrix. This is done by finding the area of the pre-image and multiplying by the matrix's determinant. Created by Sal Khan.

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Video transcript

- [Tutor] We're told to consider this matrix transformation or this is a matrix that you can view, represents a transformation on the entire coordinate plane. And then they tell us that the transformation is performed on the following rectangle. So this is the rectangle before the transformation and they say, what is the area of the image of the rectangle under this transformation? So 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. So the absolute value of the determinant would be the absolute value of 5 times 8, 5 times 8 minus 9 times 4, 9 times 4. Remember for a 2 by 2 matrix, the determinant is just this times this minus this times that. And so 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 30 is 120 plus 4 times 5 is another 20. So this is going to get us to 140 square units and we're done.