Chapter 2: Matrices and Linear Systems

Transcription

1 Chapter 2: Matrices and Linear Systems Paul Pearson

2 Outline Matrices Linear systems Row operations Inverses Determinants

3 Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers with m rows and n columns: a 11 a 12 a 1n a 21 a 22 a 2n A = (a ij ) = a m1 a m2 a mn The row index of a ij is i, and the column index of a ij is j. The set of all m n matrices with real entries is denoted M m,n (R). When m = n, this notation is shortened to M n (R).

4 Matrix addition and scalar mult. Definition Let A = (a ij ) and B = (b ij ) be m n matrices, and let α R. The operations of matrix addition and scalar multiplication are defined by adding corresponding entries and scaling all entries as follows: 1. A + B = (a ij + b ij ) 2. αa = (αa ij ) Example [ ] [ ] = = [ ]

5 Matrix addition and scalar mult. Theorem Let A, B, C M m,n (R) and α, β R. Let 0 denote the m n zero matrix A = A = A 3. α 0 = 0 4. A + B = B + A 5. (A + B) + C = A + (B + C) 6. (α + β)a = αa + βa 7. α(a + B) = αa + αb 8. (αβ)a = α(βa)

6 Matrix multiplication Definition Let A M m,n (R) and B M n,k (R). The product matrix AB is the m k matrix whose (i, j)-entry is the dot product (Row i of A) (Column j of B) Example Compute by hand: A B AB ] [ = [ rows, 3 cols 3 rows, 4 cols 2 rows, 4 cols ]

7 Matrix multiplication Definition Let A M m,n (R) and B M n,k (R). The product matrix AB is the m k matrix whose (i, j)-entry is the dot product (Row i of A) (Column j of B) Example Compute by hand: A B AB ] [ = [ rows, 3 cols 3 rows, 4 cols 2 rows, 4 cols ]

8 Matrix multiplication Example Does AB = BA? [ ] [ ] [ ] [ ] = = [ ] [ ] [ ] [ ] = = [ ] [ ] [ ] [ ] = =

9 Matrix multiplication Example Does AB = BA? Not always, so we write AB BA. [ ] [ ] [ ] [ ] = = [ ] [ ] [ ] [ ] [ ] [ ] = = [ ] [ ] [ ] [ ] [ ] [ ] = [ ] 3. 7 = DNE

10 Matrix operations Definition I n is the n n identity matrix with all diagonal entries equal to 1 and all other entries equal to 0. Theorem Let A M m,n (R), B M n,k (R) and α R. 1. A0 n k = 0 m k 2. I m A = A and AI n = A. In particular, if A is an n n matrix, then I n A = AI n = A. 3. If C M k,l (R), then A(BC) = (AB)C. 4. If C M n,k (R), then A(B + C) = AB + AC. 5. A(αB) = α(ab) = (αa)b.

11 Matrix operations If A = [ then by hand computation [ ] [ I 2 A = AI 3 = A0 3 2 = [ ] [ AI 2 = ] [ ] = = = ] [ ], [ [ [ ] = DNE ] = A, ] = A, ] = 0 2 2,

12 Matrix operations If [ ] A =, B = then by hand computation A(B + C) = = = AB + AC = = = , C = 2 1 [ ] [ ] [ ] [ ] [ ] 2 1 [ ] [ ] 2 [ 1 ] ,

13 Matrix operations Suppose Then: A = [ ] [ 1, b 1 = 1 ] [ 2, b 2 = 2 Ab 1 = Ab 2 = Ab 3 = A(b 1 b 2 b 3 ) = A((b 1 0 0) + (0 b 2 0) + (0 0 b 3 )) = ] [ 3, b 3 = 3 ].

14 Matrix operations Suppose A = [ ] [ 1, b 1 = 1 ] [ 2, b 2 = 2 Then: [ ] [ ] 3 6 Ab 1 =, Ab 7 2 =, Ab 14 3 = [ ] A(b 1 b 2 b 3 ) = [ 9 21 A((b 1 0 0) + (0 b 2 0) + (0 0 b 3 )) = In general, ] [ 3, b 3 = 3 ]. [ AB = A(b 1 b n ) = (Ab 1 Ab n ). ]. ].

15 Matrix transpose Definition 1. The transpose of the matrix A = (a ij ) is the matrix A T = (a ji ), i.e., rows and columns swap: T = T and = 2. A square matrix A such that A T = A is called a symmetric matrix. 3. A square matrix A such that A T = A is called a skew-symmetric matrix. Theorem Let A M m,n (R) and α R. 1. (A T ) T = A. 2. If B M m,n (R), then (A + B) T = A T + B T. 3. (αa) T = αa T.

16 Matrix transpose Theorem If A M m,n (R) and B M n,k (R), then (AB) T = B T A T. Proof. The (i, j) entry of (AB) T is (ab) T i,j = (ab) j,i = (row j of A ) (column i of B). The (i, j) entry of B T A T is (row i of B T ) (column j of A T ) = (column i of B ) (row j of A). Remark AB BA, but (AB) T = B T A T is always true.

17 Matrix transpose Example Suppose Then A = [ ], B = (AB) T = B T A T =

18 Matrix transpose Example Suppose A = [ ], B = Then B T A T = (AB) T = [ [ ] ] T = [ = ]. [ ].

19 Triangular and diagonal matrices Suppose A = (a ij ) is an n n square matrix. The diagonal entries of A are a 11, a 22,..., a nn. The matrix A is diagonal if all non-diagonal entries are zero. upper triangular if all entries below the diagonal are zero. lower triangular if all entries above the diagonal are zero. Example , , If a matrix is both lower and upper triangular it is 2. The transpose of a lower triangular matrix is

20 Triangular and diagonal matrices Suppose A = (a ij ) is an n n square matrix. The diagonal entries of A are a 11, a 22,..., a nn. The matrix A is diagonal if all non-diagonal entries are zero. upper triangular if all entries below the diagonal are zero. lower triangular if all entries above the diagonal are zero. Example , , If a matrix is both lower and upper triangular it is diagonal. 2. The transpose of a lower triangular matrix is an upper triangular matrix.

21 Outline Matrices Linear systems Row operations Inverses Determinants

22 Systems of Linear Equations Definition An m n system of linear equations in variables x 1, x 2,..., x n is a list of m equations of the form a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.. a m1 x 1 + a m2 x a mn x n = b m Any point (x 1, x 2,..., x n ) which satisfies all the equations in the system is called a solution of the system. A system that has at least one solution is called consistent, while a system with no solutions is called inconsistent.

23 Systems of linear equations Each graph below is the graph of a system of three linear equations in three unknowns. Determine which systems are consistent and inconsistent and the dimension of the solution set.

24 Systems of Linear Equations Definition We can represent a linear system as Ax = b: a 11 a 12 a 1n a 21 a 22 a 2n..... } a m1 a m2 {{ a mn } A x 1 x 2. x n }{{} x = b 1 b 2. b m } {{ } b Or, more succinctly, we can write the augmented matrix (A b): a 11 a 12 a 1n b 1 a 21 a 22 a 2n b a m1 a m2 a mn b m

25 Systems of linear equations Exercise 1. Find an equation for the plane P 1 in R 3 passing through the origin and parallel to 6y 2x + 3z = 100. Find an equation for the plane P 2 parallel to both 2i + 3j and 0, 1, 0 passing through the point (0, 0, 2). 2. Write the equations for P 1 and P 2 as a linear system and as an augmented matrix. 3. Solve the linear system.

26 Systems of linear equations Exercise 1. Find an equation for the plane P 1 in R 3 passing through the origin and parallel to 6y 2x + 3z = 100. Find an equation for the plane P 2 parallel to both 2i + 3j and 0, 1, 0 passing through the point (0, 0, 2). 2. Write the equations for P 1 and P 2 as a linear system and as an augmented matrix. 3. Solve the linear system. { 2x + 6y + 3z = 0, z = 2.

27 Outline Matrices Linear systems Row operations Inverses Determinants

28 Solving linear systems The linear system { x + y = 1, x + y = 4. can be solved with three operations that leave the solution set unchanged: 1. Multiplying a row by a nonzero constant (e.g., x + y = 4 2x + 2y = 8) 2. Swapping rows (order doesn t matter) 3. Adding rows together (if (x, y) satisfies both x + y = 1 and x + y = 4, then it satisfies ( x + y) + (x + y) = 1 + 4, which reduces to 2y = 5) Since each of these operations affect only the coefficients, we can store all the pertinent information about the system in an array of numbers called a matrix and record the effects of these operations in matrices as we progress toward a solution.

29 Elementary row operations Definition Elementary Row Operations (EROs) 1. αr i R i 2. R i R j 3. R i + αr j R i Definition Inverses of Elementary Row Operations 1. e : 1 α R i R i 2. e : R i R j 3. e : R i αr j R i Remark Solution sets to linear systems are unchanged by EROs.

30 Solving linear systems { 2x + y = 6 3x + 4y = 12 y L1 L2 x

31 Solving linear systems y { 2x + y = 6 3x + 4y = L2 4 L1 x y 3R 1 R 1 2R 2 R 2 { 6x + 3y = 18 6x 8y = L2 4 L1 x

32 Solving linear systems y { 2x + y = 6 3x + 4y = L2 4 L1 x y 3R 1 R 1 2R 2 R 2 { 6x + 3y = 18 6x 8y = L2 4 L1 x y R 1 + R 2 R 2 { 6x + 3y = 18 0x 5y = L2 L3 4 L1 x

33 Solving linear systems 1 3 R 1 R 1 { 2x + y = 6 0x 5y = 6 y L1 L2 L3 x

34 Solving linear systems 1 3 R 1 R 1 { 2x + y = 6 0x 5y = 6 y L1 L2 L3 x y 4 L1 L4 5R 1 + R 2 R 1 { 10x + 0y = 24 0x 5y = L2 L3 x

35 Solving linear systems 1 3 R 1 R 1 { 2x + y = 6 0x 5y = 6 y L1 L2 L3 x y 4 L1 L4 5R 1 + R 2 R 1 { 10x + 0y = 24 0x 5y = L2 L3 x y 4 L1 L R 1 R R 2 R 2 { 1x + 0y = 2.4 0x + 1y = L2 L3 x

36 Solving linear systems ( )

37 Solving linear systems ( ) 3R 1 R 1 2R 2 R ( )

38 Solving linear systems ( ) 3R 1 R 1 2R 2 R 2 R 1 + R 2 R ( 6 3 ) ( 6 3 )

39 Solving linear systems ( ) 3R 1 R 1 2R 2 R 2 R 1 + R 2 R ( ( 6 3 ) R 1 R 1 ( ) )

40 Solving linear systems ( R 1 R 1 2R 2 R 2 R 1 + R 2 R 2 ) ( 6 3 ) ( 6 3 ) R 1 R 1 ( R 1 + R 2 R 1 ( ) )

41 Solving linear systems ( R 1 R 1 2R 2 R 2 R 1 + R 2 R 2 ) ( 6 3 ) ( 6 3 ) R 1 R 1 ( R 1 + R 2 R 1 ( 10 0 ) R 1 R 1 ( 1 0 ) R 2 R )

42 EROs and elementary matrices Definition Any ERO e may be represented as a matrix E = e(i) called an elementary matrix. The result of applying the ERO e to the matrix A is the same as the matrix product EA. Example The ERO e : R 1 + 2R 2 R 1 applied to A = is the same as EA, where E = e(i 3 ) =

43 Reduced row echelon form Definition A matrix A M m,n (R) is in reduced row echelon form (rref) if: 1. The leading entry in each nonzero row is All rows of zeros occur at the bottom of the matrix. 3. Every leading 1 occurs farther to the right of the leading 1 in the previous row. 4. Every entry above and below a leading 1 is equal to 0. Example Suppose represents any real number. RREF could look like: ,

44 Gauss-Jordan Elimination Given a matrix A, its reduced row echelon form can be constructed by: 1. Starting with i = 1, find the row R j with i j whose leading entry is farthest to the left. If j i, do R i R j. 2. If the leading entry of R i is α 1, do (1/α)R i R i. This establishes a leading 1 in R i. 3. Make all entries above and below a leading 1 equal to 0 by doing EROs. 4. Go to the next row and repeat the process until the matrix is in rref.

45 Gauss-Jordan Elimination

46 Gauss-Jordan Elimination R 1 + R 2 R

47 Gauss-Jordan Elimination R 1 + R 2 R 2 3R 1 + R 3 R

48 Gauss-Jordan Elimination R 1 + R 2 R 2 3R 1 + R 3 R 3 R 2 + R 3 R

49 Gauss-Jordan Elimination R 1 + R 2 R 2 3R 1 + R 3 R 3 R 2 + R 3 R 3 R 1 2R 2 R

50 Gauss-Jordan Elimination If possible, solve the system x + 2y = 1, 2x 3y = 1, 3x + 5y = 0.

51 Gauss-Jordan Elimination If possible, solve the system x + 2y = 1, 2x 3y = 1, 3x + 5y = 0. Since rref = the system is consistent and the three given lines in R 2 intersect at the point (x, y) = ( 5, 3). There is only one solution (it is the unique solution) and it is zero dimensional.

52 Gauss-Jordan Elimination If possible, solve the system x + 2y + z = 0, 2x 3y + z = 0, 3x + 5y = 0.

53 Gauss-Jordan Elimination If possible, solve the system x + 2y + z = 0, 2x 3y + z = 0, 3x + 5y = 0. Since rref = the system is consistent and the three given planes in R 3 intersect. The first two rows say x 5z = 0 and y + 3z = 0. Since there is no pivot in the third column, z is free and z = z. Thus, the solution set is x = 5z, y = 3z, and z = z, i.e., the 1D line thru the origin (x, y, z) = (5z, 3z, z) = z 5, 3, 1 where z is any real number.

54 Reduced Row Echelon Form 1. A leading 1 in a row is also called a pivot. 2. A column without a pivot is called a free column and the variable it represents is called a free variable. The number of free variables determines the dimension of the solution set. 3. If a pivot occurs in the augmentation column, the system has no solution and is said to be inconsistent. Exercise 1. Solve Ax = 0 given A = EROs rref(a) = B = EROs rref(b) =

55 RREF and Gauss-Jordan Elimination Definition 1. A system of linear equations Ax = 0 is called a homogeneous system. 2. The solution set X H = {x Ax = 0} of a homogeneous system is called the homogeneous solution set. Theorem 1. 0 X H 2. For all x 1, x 2 X H, x 1 + x 2 X H 3. For all x X H and all α R, αx X H Remark These properties say that X H is a subspace. We ll study subspaces in chapter 3.

56 Solution sets Suppose A M m,n (R) and b R n. The solution set for a consistent homogeneous linear system Ax = 0 is X H = {x Ax = 0}. The solution set for a consistent non-homogeneous linear system Ax = b is X G = p + X H = {x Ax = b}, a translation of X H by a particular solution p (i.e., Ap = b). What happens if p X H? z p + XH = {x Ax = b} z p + XH = {x Ax = b} p + tv p p XH = {x Ax = 0} tv XH = {x Ax = 0} v y y x x

57 Systems of Linear Equations Write the system of linear equations 3x + y 3z = 2 x + y z = 0 x z = 1 as an augmented matrix (A b), solve the system by row reduction, and write the solution in the form p + X H.

58 Systems of Linear Equations Write the system of linear equations 3x + y 3z = 2 x + y z = 0 x z = 1 as an augmented matrix (A b), solve the system by row reduction, and write the solution in the form p + X H. z rref(a b) = X H p + X H v v p y x

59 Solving linear systems

60 Solving linear systems R 1 R

61 Solving linear systems R 1 R 3 R 1 + R 2 R

62 Solving linear systems R 1 R 3 R 1 + R 2 R 2 3R 1 + R 3 R

63 Solving linear systems R 1 R 3 R 1 + R 2 R 2 3R 1 + R 3 R 3 R 2 + R 3 R

64 Outline Matrices Linear systems Row operations Inverses Determinants

65 Finding inverses = A

66 Finding inverses 3R 1 + R 2 R = A = E 1 A

67 Finding inverses 3R 1 + R 2 R 2 2R 1 + R 3 R = A = E 1 A = E 2 E 1 A

68 Finding inverses 3R 1 + R 2 R 2 2R 1 + R 3 R 3 R 1 4R 3 R = A = E 1 A = E 2 E 1 A = E 3 E 2 E 1 A = EA

69 Finding inverses 3R 1 + R 2 R 2 2R 1 + R 3 R 3 R 1 4R 3 R = A = E 1 A = E 2 E 1 A = E 3 E 2 E 1 A = EA Therefore, I = EA. How do we find E = E 3 E 2 E 1?

70 Finding inverses E is the result of how the row operations transform I = I 0 0 1

71 Finding inverses E is the result of how the row operations transform I = I R 1 + R 2 R = E

72 Finding inverses E is the result of how the row operations transform I = I R 1 + R 2 R = E R 1 + R 3 R = E 2 E

73 Finding inverses E is the result of how the row operations transform I = I R 1 + R 2 R = E R 1 + R 3 R = E 2 E R 1 4R 3 R = E 3 E 2 E 1 = E 2 0 1

74 Finding Inverses Definition Given an n n matrix A with real entries, we say that A is invertible if there exists an n n matrix E with real entries such that EA = I, in which case we call E an inverse of A and write E = A 1. Theorem Given A M n,n (R), A 1 exists if there is some sequence of row operations that reduces A to the identity matrix (i.e., if rref(a) = I). Applying those same row operations to I yields E = A 1, hence applying those row operations simultaneously to A and I gives a method for finding A 1 : (A I) EROs (I A 1 ). If rref(a) I, then A 1 does not exist.

75 Using inverses Exercise 1. Given E and A below, verify that E is an inverse of A. E = , A = 2. Find a general formula for the dot product (row i of E) (column j of A). 3. Find AE. Is A an inverse of E? 4. Find a general formula for the dot product (row i of A) (column j of E)

76 Using inverses Exercise Let A = , and E = A 1 = Use E = A 1 to find the intersection of the planes x + 4z = 2, 3x + y + 12z = 1, and 2x + 7z = If b = 2, 1, 3, use E = A 1 to solve Ax = b for x. 3. If b is any vector in R 3, find the general solution to the linear system Ax = b If B = 1 10, solve AX = B If B is any matrix in M 3,k (R), find the general solution to AX = B.

77 Using inverses Exercise 1. If possible, find the inverse of A = What is the solution set X H to the homogeneous linear system Ax = 0? Is B = invertible? If so, find its inverse What is the solution set X H to the homogeneous linear system Bx = 0? 5. Conjecture a relationship between invertible matrices and homogeneous solution sets..

78 Computing inverses Theorem ( ) a b When ad bc 0, the matrix A = is invertible and its c d inverse is ( ) A 1 1 d b =. ad bc c a Example The inverse of A = A 1 = 1 5 ( ( ) is ) = ( ).

79 Properties of inverses Theorem Suppose the matrices below are all n n invertible matrices with real entries. 1. (A 1 ) 1 = A 2. (AB) 1 = B 1 A 1 3. (A 1 A 2 A k ) 1 = A 1 k A 1 2 A 1 1 Theorem Let A M n (R) be invertible. 1. If B M n,k (R) and AB = 0, then B = 0 2. If C M m,n (R) and CA = 0, then C = 0 3. If B, C M n,k (R) and AB = AC, then B = C 4. If B, C M m,n (R) and BA = CA, then B = C

80 Using inverses Exercise Solve A T + XB = C, where A = ( ) ( 1 1, B = 3 2 ), C =

81 Using inverses Exercise Solve A T + XB = C, where A = ( ) ( 1 1, B = 3 2 ), C = Since XB = C A T, it follows that X = (C A T )B 1 and thus X =

82 Outline Matrices Linear systems Row operations Inverses Determinants

83 Computing determinants Definition ( a b Let A = c d ). Then det(a) = A = ad bc. Theorem Let A = (a ij ), and let A ij denote the ij-minor of A obtained by deleting the ith row and jth column of A. Then det(a) = ( 1) i+j a ij det A ij where the sum is taken from j = 1 to j = n if expanding along row i and from i = 1 to i = n if expanding along column j. The quantities det(a ij ) are (n 1) (n 1) determinants, and this process can be repeated until only determinants of 2 2 matrices remain.

84 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

85 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

86 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

87 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

88 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

89 Computing determinants Alternating signs: Minors of A = a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a det(a)=+a 2,2 a 2,3 1,1 a 3,2 a 3,3 a 1,2 a 2,1 a 2,3 a 3,1 a 3,3 + a 1,3 a 2,1 a 2,2 a 3,1 a 3,2 = a 1,2 a 2,1 a 2,3 a 3,1 a 3, a 2,2 a 1,1 a 1,3 a 3,1 a 3,3 a 3,2 a 1,1 a 1,3 a 2,1 a 2,3

90 Computing determinants Theorem Let A M n,n (R). Then the value of det(a) is the same regardless of the choice of row or column to expand over at each stage of the algorithm. Theorem Given A M n,n (R), det(a) = det(a T ). Proof. (Rough sketch:) Calculating det(a) by expanding along the top row of A is exactly the same as calculating det(a T ) by expanding along the leftmost column of A T.

91 Computing determinants Example Compute det Exercise Compute det

92 Computing determinants Example Compute det = 20 Exercise Compute det = 3

93 Determinants geometrically x 2 x Ae 2 e A = 2 x x 1 e 1 Ae 1 ( ) 0 0 = ( 0 ) 1 e 1 = ( 0 ) 0 e 2 = 1( ) 1 e 1 + e 2 = 1 ( 2 0 det 0 3 ) = 6 ( ) 0 A0 = ( 0 ) 2 Ae 1 = ( 0 ) 0 Ae 2 = 3( ) 2 A(e 1 + e 2) = 3

94 Determinants geometrically x 2 x Ae 2 e A = 2 x x 1 e 1 Ae 1 ( ) 0 0 = ( 0 ) 1 e 1 = ( 0 ) 0 e 2 = 1( ) 1 e 1 + e 2 = 1 ( 2 0 det 0 3 ) = 6 ( ) 0 A0 = ( 0 ) 2 Ae 1 = ( 0) 0 Ae 2 = 3( ) 2 A(e 1 + e 2) = 3

95 Determinants geometrically x 2 x e A = 2 x Ae 1 x 1 e 1 Ae 2 ( ) 0 0 = ( 0 ) 1 e 1 = ( 0 ) 0 e 2 = 1( ) 1 e 1 + e 2 = 1 ( 2 0 det 0 3 ) = 6 ( ) 0 A0 = ( 0 ) 2 Ae 1 = ( 0 ) 0 Ae 2 = 3 ( ) 2 A(e 1 + e 2) = 3

96 Determinants geometrically x 2 x e A = 2 x Ae 1 x 1 e 1 Ae 2 ( ) 0 0 = ( 0 ) 1 e 1 = ( 0 ) 0 e 2 = 1( ) 1 e 1 + e 2 = 1 ( 2 0 det 0 3 ) = 6 ( ) 0 A0 = ( 0 ) 2 Ae 1 = ( 0 ) 0 Ae 2 = 3 ( ) 2 A(e 1 + e 2) = 3

97 Determinants geometrically e 2 e 1 A = a c = (a b d 1 a 2 ) Ae2 = a2 = ( ) c d n = ( ) b a p Ae1 = a1 = ( ) a b Note that n = b, a is perpendicular to a 1 = a, b. The signed length of p is the component of a 2 along n, comp n a 2 = b, a c, d b, a so the signed area of the parallelogram is = ad bc b, a, ad bc Signed area = (base)(height) = a, b = ad bc. b, a

98 Determinants geometrically z z x e 3 A = (a 1 a 2 a 3 ) e 1 e 2 y = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 x Ae 3 = a 3 Ae 2 = a 2 y Ae 1 = a 1 Using geometry, projections, and properties of, we showed that the triple scalar product a 1 (a 2 a 3 ) is the volume the parallelepiped formed by a 1, a 2, and a 3. It is easy to verify that det(a) = a 1 (a 2 a 3 ), so Signed volume = det(a) = a 1 (a 2 a 3 ). The linear function f : R 3 R 3 defined by f (x) = Ax has magnification factor det(a), i.e., one cubic unit of volume gets mapped to det(a) cubic units of signed volume.

99 Determinants geometrically More generally, a matrix A M n,n (R) defines a linear function f : R n R n by f (x) = Ax, and the determinant of A is the signed magnification factor for f : det(a) = signed area (or volume) of output region signed area (or volume) of input region, where we use area when n = 2, volume when n = 3, and hypervolume when n 4. The sign of det(a) is determined by the number of reflections in the linear transformation (e.g., if a right-handed coordinate system in the input space gets mapped to a left-handed coordinate system in the output space, then the sign of det(a) is negative).

100 Geometry of determinants x 2 x 2 Ae1 x 2 A 2 e 2 e 2 x 1 A x 1 e 1 A A 2 e 1 x 1 Ae 2 Suppose f (x) = Ax. 1. Find a formula for f (x). 2. Describe how f and f 2 transform the unit square (e.g., dilation, reflection, rotation, shear, projection, etc.). 3. Find the signed magnification factor for f, f 2, f 3, and f k. 4. Find a formula for f 1, if possible. 5. Find the signed magnification factor for f 1 and f k. What is det(a 1 )?

101 Geometry of determinants x 2 x 2 x 2 Ae 2 A 2 e 2 e 2 x 1 A x 1 A x 1 e 1 Ae 1 A 2 e 1 Suppose f (x) = Ax. 1. Find a formula for f (x). 2. Describe how f and f 2 transform the unit square (e.g., dilation, reflection, rotation, shear, projection, etc.). 3. Find the signed magnification factor for f, f 2, f 3, and f k. 4. Find a formula for f 1, if possible. 5. Find the signed magnification factor for f 1 and f k. What is det(a 1 )?

102 Geometry of determinants 1. Find a nonzero 2 2 matrix A = (a 1 a 2 ) with real entries such that det(a) = 0. What is the geometric relationship between the column vectors a 1 and a 2? 2. If A = (a 1 a 2 a 3 ) is in M 3,3 (R) and det(a) = 0, what is the geometric relationship among the column vectors a 1, a 2, and a 3? 3. Suppose a linear function f : R 3 R 3 defined by f (x) = Ax for some A M 3,3 (R) maps a sphere of radius 3 in the input space to an ellipsoid with volume 7 in the output space. What can you say about the determinant of A?

103 Geometry of determinants 1. Suppose A = , B = Since det(a) = 3, what is det(b)? 2. Suppose A = (a 1 a 2 a 3 ) is in M 3,3 (R) has det(a) = 3. What are det(a 2 a 1 a 3 ), det(a 2 a 3 a 1 ), det(2a 1 5a 2 a 3 ), and det(5a)? 3. Find the area of the hexagon with side length 1.

104 Properties of determinants Theorem Let A, B M n,n (R). 1. det(a) = det(a T ). 2. det(ab) = (deta)(detb). 3. A is invertible if and only if det(a) 0, in which case det(a 1 ) = 1 det(a). 4. If A = (a ij ) is an upper-triangular, lower-triangular, or diagonal matrix, then det(a) = a 11 a 22 a nn. 5. If A has a row or column of zeros, then det(a) = For all n 1, det(i n ) = 1.