EP2.2/H3.1. Higher-Order Determinants The 1 1 matrix [a] [ is ] invertible exactly when a 0. The 2 2 matrix is invertible exactly when ad bc 0.

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1 EP22/H31 Higher-Order Determinants The 1 1 matrix [a] [ is ] invertible exactly when a 0 a b The 2 2 matrix is invertible exactly when c d ad bc 0 What about a 3 3 matrix? Is there some short of expression which will determine when such a matrix is invertible? The answer is yes (and the answer for larger square matrices is also yes), and it is called the determinant of the matrix, but the formula gets very complicated very fast as the matrix gets bigger; it is easier to work on an individual basis

2 Let s proceed one step at a time, starting with the minors of a matrix A The determinant of the matrix obtained by removing all the entries in the ith row and the jth column of A is called the (i, j)th minor of A and is denoted M i,j This assumes you know how to find the determinant of a smaller matrix!

3 If A = , then M 1,2, (1, 2)th minor of A, is the determinant of [ ] =, which is ( 3) = M 2,1 is the determinant of [ ] =, which is ( 4) 1 ( 6) 3 =

4 The (i, j) cofactor of A, denoted C i,j, is ( 1) i+j M i,j The ( 1) i+j creates a checkerboard pattern, which changes the signs of some of the minors:

5 The (i, j) cofactor of A, denoted C i,j, is ( 1) i+j M i,j The ( 1) i+j creates a checkerboard pattern, which changes the signs of some of the minors: Minors

6 The (i, j) cofactor of A, denoted C i,j, is ( 1) i+j M i,j The ( 1) i+j creates a checkerboard pattern, which changes the signs of some of the minors: Minors Cofactors

7 The (i, j) cofactor of A, denoted C i,j, is ( 1) i+j M i,j The ( 1) i+j creates a checkerboard pattern, which changes the signs of some of the minors: Minors Cofactors

8 The (i, j) cofactor of A, denoted C i,j, is ( 1) i+j M i,j The ( 1) i+j creates a checkerboard pattern, which changes the signs of some of the minors: Minors Cofactors

9 Now (at last) we can find the determinant of A, which is defined to be A 1,1 C 1,1 + A 1,2 C 1,2 + A 1,3 C 1,3 + + A 1,n C 1,n The determinant of a matrix A is denoted by A or det A Here, the determinant is (2)( 26) + ( 4)( 13) + (3)(0) = 0 This method is called Expansion by Minors

10 A more interesting example, with some of the details omitted, is: = = 2 ( ) 7 ( ) = ( )

11 As you can see, calculating the determinant of a 3 3 matrix requires calculating the determinant of matrices Calculating the determinant of a 4 4 matrix requires calculating the determinant of matrices, each of which requires the determinants of matrices This makes a total of matrices Calculating the determinant of a 5 5 matrix requires calculating the determinant of matrices, which will require the determinant of matrices

12 In general, the determinant of an n n matrix requires the determinants of 1 n! 2 2 matrices This 2 is not an efficient procedure! (n! = 1 2 n 10! is around 3 million, 70! is bigger than a googol) So how can we cut down the computation time?

13 First of all, we can expand along any row and get the same answer, not just the first one Also, we can expand along any column So if some row or column has a lot of 0 s in it, we can cut down the number of computations We need to obey the checkerboard pattern, so the first determinant might be subtracted instead of being added

14

15

16 = Remember:

17 =

18 =

19 = ( = ) Remember: + + +

20 = =

21 = = = 2 4 ( ) = 72 Only one 2 2 determinant had to be calculated here!

22 Now suppose we have an upper triangular matrix* Maybe we want to find the determinant of: π * An upper triangular matrix is a matrix A where A i,j = 0 whenever j < i

23 π =

24 π =

25 π =

26 π =

27 π = Where do these numbers come from?

28 π = The determinant of an upper (or lower) triangular matrix is the product of the entries on the diagonal Because Gaussian Elimination puts a matrix into upper triangular form, you can use row operations to get the matrix into a form where you can calculate the determinant easily If you do this sort of calculation, you also need to keep track of what effects the row operations have on the determinant However, we will not do this in this class

29 Some useful properties of determinants are the following: det A 0 exactly when A has an inverse det(ab) = det A det B det(a 1 ) = 1 det A det(ra) = r n det A if A is n n det(a + B) has no nice formula

30 And now for the shortcut for the determinant of a 3 3 matrix, known as Sarrus's Method This will NOT work for larger matrices!!! Suppose you want to find the determinant of

31 First, rewrite the first two colums to the right of the matrix (to make the shortcut easier to remember):

32 There are six diagonals of this matrix Three of them are called forward diagonals and are shown below

33 The other three are called backwards diagonals

34 Multiply the entries along each of the six diagonals & & &

35 Now, you add the products you get from the forward diagonals, and subtract the products you get from the backwards diagonals to get the determinant: & & &

36 & & & So = (54) + (42) + (21) (27) (28) (63) =

37 Exercise For which value(s) of x is det A = 0, where A = x + 2 1? 0 4 x 1

38 x x x 1 0 4

39 x x x det A = +(1)(x + 2)(x 1) + (2)( 1)(0) + (0)( 1)(4) (0)(x + 2)(0) (1)( 1)(4) (2)( 1)(x 1) = (x + 2)(x 1) + (4) + (2)(x 1) = x 2 + x x 2 = x 2 + 3x = x(x + 3) Then det A is zero when x is 0 or 3

40 Test 1: The List For Test 1, you will need to know how to: Solve systems of linear equations using Gaussian Elimination and Gauss-Jordan Elimination [12, 13] Determine how many solutions a system of linear equations has, if the matrix is in row echelon form [13] Describe the solutions if a system of linear equations has infinitely many solutions [13] Do matrix arithmetic (including solving equations) [14]

41 Test 1: The List (Continued) Find the inverse of a matrix [15] Use the inverse formula X = A 1 B for solving a system of linear equations [15] Calculate determinants using Expansion by Minors and possibly Sarrus s Method [22] Remember: You will not be allowed to use a graphing calculator, or any other calculator, for that matter!