Math 110 Fall 05 Lectures notes 24 Oct 24 Monday Now we begin Chapter 4 determinants Determinants are useful in several fields of mathematics Linear Algebra deciding if A is invertible defining eigenvalues Geometry finding volumes of parallelograms in 2D or parallelepipeds in any dimension Calculus changing variables in a multiple integral There are several equivalent definitions useful in different situations all derivable from one another We start with n 1 and n 2 for which some definition look way too complicated and then see that for higher dimensions they work best We will assume the field F does not have characteristic 2 when needed The determinant of a 1x1 matrix A a is just a The determinant of a 2x2 matrix A x1 y1 is x2 y2 1 Explicit formula det A x1 y2 y1 x2 2 Recursive formula det A x1 det y2 y1 det x2 This is identical to the explicit formula in the 2x2 case but will extend to larger n 3 Oriented area of a parallelogram Let P be the parallelogram with 3 corners at 0 0 x1 y1 and x2 y2 This means the 4th corner must be x1 x2 y1 y2 picture Recall area of parallelogram Base x height ASK WAIT Why Consequence can take one side slide it parallel to other side without changing area For example we could replace x2 y2 by x2 y2 c x1 y1 for any c without changing area Let s pick c to make it easy to figure out base and height picture First slide top edge to put corner on y axis i e pick c so x2 y2 c x1 y1 0 y for some y Thus c x2 x1 assume x1 nonzero for the moment and y y2 x2 x1 y1 Second slide right edge to put corner on x axis i e pick c so x1 y1 c 0 y x1 0 We see we get a rectangle with the same area as the parallelogram 1 area base height x1 y2 x2 x1 y1 x1 y2 x2 y1 We call this the oriented area because it could be negative Its absolute value is the usual area If x1 is zero we don t have to do one of the slides and end up with the same answer The orientation is easy to understand geometrically in this 2 by 2 case If moving from side 1 to side 2 within the parallelogram means you move counterclockwise the orientation 1 positive other it is 1 In high dimensions it is harder to explain geometrically which is why we use the other algebraic definitions 4 LU factorization Assuming x1 is nonzero we can do the LU factorization A x1 y1 1 0 x1 y1 L U x2 y2 x2 x1 1 0 y2 x2 x1 y1 and just take the product of the diagonal entries of U x1 y2 x2 x1 y1 x1 y2 x2 y1 Note that the diagonal entries of U are the same numbers we get from sliding edges This is not a coincidence We will later generalize this to the case A P L L U P R 5 Axiomatic definition The determinant of an n x n matrix is the function det M n x n F F satisfying 1 det A is a linear function of each row or column In other words if A x is a matrix with row i equal to x and the other rows fixed then det A c x y c det A x det A y 2 swapping two rows or columns of A changes the sign of det A provides orientation 3 det I 1 obvious volume of unit cube To illustrate axiom 1 in the 2 x 2 case det c x1 d x1 c y1 d y1 x2 y2 c det x1 y1 d det x1 y1 x2 y2 x2 y2 So even though the determinant itself is a polynomial it is actually a linear function of any row or column which we will find very useful One might ask why oriented area instead of just area I e why not take absolute values in all these definitions Because then we would 2 lose the linearity property just described which we will need So if you want the usual area or volume just take the absolute value at the end 6 Product of A s eigenvalues But we haven t defined eigenvalues yet Now let s look at the definitions for n 2 1 Explicit formula Written out it would be a polynomial of degree n with n terms n grows quickly 3 6 4 24 5 120 10 362880 so this is not so useful as later definitions 2 Recursive formula This is the starting definition used by the textbook Def Let A be an n by n matrix Then A tilde ij is the n 1 by n 1 matrix gotten by deleting row i and column j of A Recursive definition of Determinant If A a is 1 by 1 det A a Otherwise det A sum j 1 to n 1 1 j A 1j det A tilde 1j A 11 det A tilde 11 A 12 det A tilde 12 A 13 det A tilde 13 3 Oriented volume of a parallelepiped In the 3 by 3 case think of the parallelepiped P with corner at the origin and the points defined by the 3 rows of A Altogether A has 8 corners whose coordinates are gotten by taking summing all possible subset of the 2 3 8 rows of A picture P s volume with an appropriate orientation or sign is det A In the n by n case P will also have corners at the origin and the n points defined by the n rows of A Altogether A has 2 n corners gotten from summing all possible subsets of A s rows Again P s volume with an appropriate orientation is det A The easiest way to see this is from the other definitions and as in the 2 by 2 case interpreting them as changing the parallelepiped sliding edges to another one with the same volume and all perpendicular edges a box whose volume is just the product of the edge lengths 4 LU factorization Using A P L L U P R will be the best way to actually compute det A in practice for large matrices det A 0 if rank A n det P L det P R U 11 U 22 U nn if rank A n 3 where det P L and det P R are both either 1 or 1 and easy to figure out We will return to this once we understand the other definitions 5 Axiomatic Definition This is same as above The determinant of an n x n matrix is the function det M n x n F F satisfying 1 det A is a linear function of each row 2 swapping two rows of A changes the sign of det A 3 det I 1 Our next goal is to how that the recursive formula satisfies all these properties of …