Continuing our discussion of algebra and linear algebra, now I want to talk about matrices and determinants. So, a matrix as defined in the reference handbook is an ordered set of elements arranged in a rectangles of M rows and N columns and set off in brackets, as shown in the extract from the reference handbook here. And, in this case, the first one here, the matrix A has three rows and two columns. And B is two rows by two columns, and the number of manipulations that we can perform on matrices, for example, multiplying them together. And as stated here, for multiplication to be possible, The number of columns in the first matrix A must be equal to the number of rows in B. So in this example here, C is the product of the matrices A and B. And the general rule here is that the first element here, this element, is equal to A times H plus B times J. Etc. for the other elements. And the second one here, this one, is A x I + B x K, etc. are the rules for multiplying matrices together. Additional matrices shown here is simpler, and additional matrices, the number of rows and columns in both matrices must, of course, be the same. And here we simply add the corresponding elements. So the first element here is just A + G, etc. There are a couple of other things, which I mentioned, which probably won't arise, but the identity matrix is a square matrix. In other words, n by n, where all of the diagonal values are one, and all of the other elements, which are off diagonal, are zero. To transpose a matrix, we simply reverse. The rows become columns, and the columns become rows, and the transpose matrix we usually denote by a superscript t. And finally, we have the inverse of a matrix a to the minus one, is the adjoin of the matrix divided by The determinant of the matrix, but again, I give these definitions for completeness, but I doubt that they will arise in the actual exam. So, let's do some examples. We have two matrices, A and B, which are given here, and the first question is, what is the sum of the two matrices? So, to sum them up, we simply add up the corresponding elements. So the first element here is two plus one, this element seven is three plus four, etcetera. So, here is the sum of those two matrices. Next, what is the product of the two matrices. So, here we have the product, A multiplied by B, is 2 3 4 1 multiplied by et cetera, and to compute this, we multiply the elements of each row by the elements of the corresponding column in the second matrix. So, in this case, the first element here is 2 times 1. Plus 3 x 3 is this element. And the second element here is 2 x 4 + 3 x 0. And similarly for the bottom row. So computing all those numbers out, the answer is 11, 8, 7, 16, is the product of those two matrices. Now related to this are determinants. And a determinant is a scalar quantity which is computed from a square matrix. In other words, the number of rows and columns Is the same, and the definitions and the computations from the handbook are given over on the right-hand side here. So, for example, a determinant, we write either this way, Det(A) or A confined within two Vertical lines, and the simplest case is a 2-by-2 matrix, a1, a2, b1, b2. And the value of the matrix is the crossproducts here a1, b2 minus a2, b1 is the value of the determinant of that matrix, which is the equation given right here. For a third order determinant or a 3 by 3, determinant of a matrix, a1, a2, a3, et cetera shown here, the answers a little bit more complicated. The result is given here And an easy way to compute the value of the determinant is to compute the values one at a time. So first of all, I take a1 here, and cross out the row and column containing that. So I have a2 multiplied by b2, c3. Minus b3, c2, which is the first term here. Next, I go to the coefficient a2, again, cross out that row and column, and the signs alternate when I do this. Is plus, minus, plus. So this is going to be minus a2 multiplied by b1, c3 minus b3, c1, etcetera. And I could do that for the last one here, so that's a simpler way to remember and to compute determinants. And, if you expand these out, I suggest you check that you can show that those two equations are the same. So, let's do a couple of examples on that. Firstly, the determinant of this two by two matrix is most nearly which of these alternatives? So, this one is very simple. It's just 2 times 3, minus 4 times 1, and the answer is two, so the answer is A. But a three by three matrix is a little bit more complicated. So, the determinant of this three by three matrix Is most nearly which of these. So, in this case, again I follow the rule that I just gave you. So start off with the number 2 here. Cross out the row and column and that is equal to 0 times 1 minus 2 times 4, which is this term here. Next I go to 3 And it’s minus 3 because the signs alternate. Again, cross out the column and row, and I form 1x1-2x3, which is this term, and finally, the last term here, 1, cross out the row and column, it’s plus because, again, the sign has alternated, plus 1. Multiplied by 1 x 4- 0 x 3. And expanding that out and evaluating it, the answer is 35, so the answer is C. And this concludes our discussion of matrices and determinants.
