Welcome. For this lesson, we will review matrix arithmetic. The outcomes are here. We'll review how to add matrices, how to multiply matrices by scalars, how to multiply matrices and matrix multiplication rules. In previous lectures, we learned how to add matrices, how to multiply matrices by scalars, and how to multiply matrices. In this lesson we will review those operations and we'll do this by looking at some examples. The first example is this, let's write 2A times u plus v as a single matrix, where a is the matrix 2, 1, 0, minus 1, u is the column vector 4, minus 1, and v is the column vector minus 3, 5. Well, we can see from this that 2A is just multiplying each entry of a by 2. That gives me 4, 2, 0, minus 8. Now if I look at u plus v, all I'm doing is adding the corresponding entries, so 4 plus or minus 3 is 1, minus 1 plus 5 is 4. If you remember what I showed is that 2 A was 4, 2, 0, minus 1 and u plus v is 1, 4, so I want to multiply these. Now remember this, here is a 2 by 2 matrix. This is a 2 by 1, so the product will be a 2 by 1 which we see we have here. I take this row times this column that gives me 4 plus 8 which gives me 12. If I take this row times this column, I have a minus 8 times 4 which gives me a minus 32. The next example says I have these 2, 3 by 3 matrices A and B. I want to find and simplify the first is B times A inverse B inverse and a second, whereas A transpose minus B and that whole matrix sum transpose. I want to make a few remarks first before I go through and work this problem. The first one is that, if you remember, a matrix is invertible if and only if its determinant is not equal to zero. If A and B are invertible matrices of the same size and A times B is also invertible and if I look at A and B inverse, B inverse times A inverse, so it reverses the order of the product. If A is invertible then the inverse of A inverse is A. Final note that all bakers about transpose and A transpose transpose is A. The other one, if I take A plus B transpose that gives me A transpose plus B transpose. Now let's go look at our solution for this B times A inverse B inverse. First thing I want to do is it makes sure that these inverses makes sense. Here I have the determinant of A and I see that it's 12 and determinant of B is actually a minus 20. That tells me then that both A inverse and B inverse exists. That means what I can do when I look at this equation, B A inverse, B inverse, all I'm going to do is use these rules first and it will make this a really easy problem to do. When I take the inverse of this product here, remember it reverses it so that gives me the B inverse and I have A inverse inverse. B times B inverse is just the identity I. A inverse inverse is just A. This product here is, this is the identity is just A, so the answer to this one really is just A. There was not much you had to actually do on this if you knew the rules. The second one that I have is A transpose minus B, the whole thing transpose. If I look at this, remember the sum is the sum of the transpose. This will be A transpose transpose minus B transpose and A transpose transpose is just A. Here I have A which is 1, 0, 1, 0, 3, 0, 1, 0, 5, and here I have B which is minus 1, 0, 1, 0, 5, 6, 3, 0, 1 transpose. I want to take the transpose of B. Remember when you take the transpose, your rows become columns. This here is my first row here so becomes my column. Here's the second row, becomes the second column and this row here becomes this column this is B transpose here. What I'm looking at here then is A minus B transpose, this is a transpose of B. Now I just sum these up by adding the corresponding entries, so 1 minus minus 1 gives me 2. Here I have 0 minus 0. Here I have 1 minus 3. That gives me minus 2, 0 Minus 0 is 0, 3 minus 5 gives me a minus 2. I get zero here again. 1 minus 1 is 0. This is 0 minus 6, so it gives me minus 6 and 5 minus 1 is 4,. In this lesson we learned some rules for the transpose of a matrix and for the inverse of a matrix. All these rules are easily proved that can be found in most linear algebra textbooks.
