The Identity Matrix
When dealing with matrix computation, it is important to understand the identity matrix. We can think of the identity matrix as the multiplicative identity of square matrices, or the one of square matrices. Any square matrix multiplied by the identity matrix of equal dimensions on the left or the right doesn't change. The identity matrix is used often in proofs, and when computing the inverse of a matrix.
We're talking about square matrices and one really important square matrix is the identity matrix we'll talk about that in a second. First, let's take a look at matrix a, b and i, I first want to multiply a times i so going across 1 times 1 is 1 plus 0 is 1, 0-1 is -1, 3+0 is 3 and 0+0 is 0, so notice 1,-1,3,0 1,-1,3,0 I get the same matrix a back from this from this product, this is actually just matrix a.
And now let me multiply i times b, so here's i and here's b I get 2+0 2, here 3+0 3, I get 0-1 -1 and I get 0+4 4 and again 2,3 -1,4 exactly the same, this is matrix b so what we're noticing here is that when we multiply by this special matrix i, we get the matrix we started with so 8 times i will just be a, i times b will just be b, i is called the identity matrix. It's the identity matrix of order 2, so an identity matrix is a matrix that has ones down the diagonal and everywhere else it has zeros. Others other orders of square matrices have them too. Here's the identity matrix of order 3, here's the identity matrix of order 4 and just like you saw before if I multiply a 3 by 3 matrix by this matrix it will remain unchanged so these are kind of like the multiplicate of identity of real numbers which is the number 1 you multiply 5 times 1 you get 5 so the multiplication leaves the number unchanged.
The identity matrix is really important in the Algebra of matrices as you'll see in a coming episode.