Want to talk about square matrices. We're going to be talking a lot about square matrices in the next few lessons.
A square matrix is a matrix with the same number of rows and columns. Like these two. These are both square matrices of order 2. The order is just a number of rows and columns. Let's multiply a times b. Square matrices multiply just like regular matrices. I'm going to go 2 plus 1, 3. 3 minus 4, -1. 6 plus 0, 6. 9 plus 0, 9.
Let's first observe that the product of an order of 2 order 2 matrices is another order 2 matrix. 2 by 2 times a 2 by 2 is still a 2 by 2. We just multiplied a times b. Let's see what we get when we multiply b times a. I get 2 plus 9, 11. I get -2 plus 0. I get -1 plus 12. I get 1 plus 0. Right off the bat I also want to notice that this matrix, b times a, is not the same as a times b.
We see that not only should we expect to get a 2 by 2 matrix, whenever we multiply 2 by 2 matrices. Matrix multiplication is not commutative. a times b does not in general equal b times a. I should just say in general because sometimes it does. but in general matrix multiplication is not commutative.