Unit
Systems of Linear Equations and Matrices
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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.