Composing 3x3 matrices | Matrices | Precalculus | Khan Academy

So, we have two three by three matrices here: matrix A and matrix B. We could, of course, view each of them as a transformation in three-dimensional space.

Now, what we're going to think about in this video is the composition of A and B. So, you could think of this as the transformation where you apply B first, and then you apply A after that. Then, we can represent that by another 3x3 matrix, which is partially completed here. We have the first and the third column here, and so my question to you is: what is this middle column where I have these three blanks? Pause the video and try to work through that.

Alright, now let's work through this together. One way to think about how to construct A of B is that what you're doing is you're taking each of the columns of B and you're thinking about what they would be under the transformation A. So, if you were to apply the transformation A to this column right over here, you would get this column. If you apply the transformation A to this column right over here, you would get this column.

So, what we really need to do is apply the transformation A to this column, to the middle column right over here. And just as a reminder, how this transformation works: a vector (0, 2, 3) you can think of it as 0 of the (1, 0, 0) vector, the unit vector in the x direction, plus two of the (0, 1, 0) vector plus three of the (0, 0, 1) vector.

Now, when you apply the transformation, instead of using these unit vectors, you use the image of them under this transformation. And now, in this situation, instead of a (1, 0, 0) vector, we are going to be using this thing. Instead of a (0, 1, 0), we're going to be using this thing. Instead of a (0, 0, 1), we are going to be using this thing.

So, this middle column, what is transformed by this vector, is going to be zero instead of the (1, 0, 0) one. It's going to be zero of the (-3, -3, 3) vector, and then we have plus two, plus two of the zero let me do that in that purple color, of the (0, -2, 3) vector. Then, last but not least, you're going to have three of the plus three of them—I’ll do that in yellow—the (0, -4, 1) vector.

Now, we just quote do the math. So, when you apply zero times all of this, you're just going to have a (0, 0, 0) vector. So, we can let those all go away, and then you are left with, let’s see, this one is going to be 2 times 0 is 0. 2 times -2 is -4. 2 times 3 is 6. You're going to have that plus 3 times 0 is 0. 3 times -4 is -12. 3 times 1 is 3.

I could have written this a little bit neater but hopefully you get the idea. And then, when we add those two things: 0 plus 0 is 0. -4 plus -12 is -16. 6 plus 3 is 9. And we're done! We have just completed the composition of A of B.