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.
Let's multiply some 3 by 3 matrices. I've got three of them here; a, b and a special one i. Notice that i has 1's down the diagonal and everything else is 0.
Let's compute 8 times i. Remember we go across the row of the left matrix, down the column of the right. I'm going to get 3 plus 0 plus 0, 3. I'm going to get 0 plus 2 plus 0, 2. I'm going to get 0 plus 0 plus 6, 6. 1 plus 0 plus 0, 1. 0 plus 1 plus 0, 1. 0 plus 0 plus 2, 2. 2 plus 0 plus 0, 0 plus 2 plus 0 and finally 0 plus 0 plus 5. Notice that multiplying matrix a by this matrix i, give us the very same matrix a. This is matrix a all over again. This is called the identity matrix.
What makes the identity matrix special, is if you multiply it by another square matrix of the same order you get that matrix back again. It's like multiplying a real number by 1. We call this the identity matrix.
Let's take a look at another product a times b. We get 3 minus 2 plus 0, 1. We get 6 plus 6 minus 12, 0. -6 plus 0 plus 6, 0. We get 1 minus 1 plus 0, 0. 2 plus 3, 5 minus 4 is 1. I get -2 plus 0 plus 2, 0. I get 2 minus 2 plus 0, 0. 4 plus 6 minus 10, 0. Finally, -4 plus 0 plus 5, -4 plus 5,1. Notice this is the identity matrix.
This is very interesting. This was matrix a and this was matrix b and I multiplied a by b and I got the identity matrix. We call be the inverse of a, it's kind of like multiplying a number by it's reciprocal. When you multiply 5 by it's reciprocal 1/5 you get 1. Which is the multiplicative identity of real numbers.
When you multiply a matrix by its inverse you get the multiplicative identity of matrix multiplication i.
Again this is the inverse matrix of a.