Elementary Matrices, Matrix Inverse, Determinants, More on Linear - PowerPoint PPT Presentation

DM559 Linear and Integer Programming Lecture 4 Elementary Matrices, Matrix Inverse, Determinants, More on Linear Systems Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

Elementary Matrices Matrix Inverse Determinants Outline Cramer’s rule 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Cramer’s rule 2

Elementary Matrices Matrix Inverse Determinants Outline Cramer’s rule 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Cramer’s rule 3

Elementary Matrices Matrix Inverse Determinants Row Operations Revisited Cramer’s rule Let’s examine the process of applying the elementary row operations: → −     a 12 · · · a 1 n a 1 a 11 − → a 22 · · · a 2 n a 2 a 21     A =  =  . . ... .   .  . . . .     . . . .    → − a m 1 a m 2 · · · a mn a m ( − → a i row i th of matrix A ) Then the three operations can be described as: → − − → → −       a 1 a 2 a 1 λ − → − → → − a 2 + λ − → a 2 a 1 a 1        .   .   .  . . .       . . .       → − → − − → a m a m a m 4

Elementary Matrices Matrix Inverse Determinants Cramer’s rule For any n × n matrices A and B : − →  a 11 a 12 · · · a 1 n   b 11 b 12 · · · b 1 n   a 1 B  − → a 21 a 22 · · · a 2 n b 21 b 22 · · · b 2 n a 2 B       AB =  =  . . .   . . .   .  ... ... . . . . . . .       . . . . . . .      − → a n 1 a n 2 · · · a nn b n 1 b n 2 · · · b nn a n B → − → − → −       a 1 B a 1 B a 1 − → a 2 B + λ − → ( − → a 2 + λ − → → − a 2 + λ − → a 1 ) B a 1 B a 1        =  =  B  .   .   .  . . .       . . .    − → → − → − a n B a n B a n ( matrix obtained by a row operation on AB ) = ( matrix obtained by a row operation on A ) B ( matrix obtained by a row operation on B ) = ( matrix obtained by a row operation on I ) B 5

Elementary Matrices Matrix Inverse Determinants Elementary matrix Cramer’s rule Definition (Elementary matrix) An elementary matrix, E , is an n × n matrix obtained by doing exactly one row operation on the n × n identity matrix, I . Example:       1 0 0 0 1 0 1 0 0 0 3 0 1 0 0 4 1 0       0 0 1 0 0 1 0 0 1 6

Elementary Matrices Matrix Inverse Determinants Cramer’s rule  1 2 4   1 2 4  ii − i 1 3 6 0 1 2 B = − − →     − 1 0 1 − 1 0 1     1 0 0 1 0 0 ii − i  = E 1 I = 0 1 0 − − → − 1 1 0    0 0 1 0 0 1       1 0 0 1 2 4 1 2 4  = E 1 B = − 1 1 0 1 3 6 0 1 2      0 0 1 − 1 0 1 − 1 0 1 7

Elementary Matrices Matrix Inverse Determinants Outline Cramer’s rule 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Cramer’s rule 8

Elementary Matrices Matrix Inverse Determinants Matrix Inverse Cramer’s rule The three elementary row operations are trivially invertible. Theorem Any elementary matrix is invertible, and the inverse is also an elementary matrix       1 0 0 1 2 4 1 2 4  = E 1 B = − 1 1 0 1 3 6 0 1 2      0 0 1 − 1 0 1 − 1 0 1  1 0 0   1 2 4   1 2 4   =  = B E − 1 1 1 0 0 1 2 1 3 6 1 ( E 1 B ) =     0 0 1 − 1 0 1 − 1 0 1 9

Elementary Matrices Matrix Inverse Determinants Row equivalence Cramer’s rule Definition (Row equivalence) If two matrices A and B are m × n matrices, we say that A is row equivalent to B if and only if there is a sequence of elementary row operations to transform A to B . This equivalence relation satisfies three properties: • reflexive: A ∼ A • symmetric: A ∼ B = ⇒ B ∼ A • transitive: A ∼ B and B ∼ C = ⇒ A ∼ C Theorem Every matrix is row equivalent to a matrix in reduced row echelon form 10

Elementary Matrices Matrix Inverse Determinants Invertible Matrices Cramer’s rule Theorem If A is an n × n matrix, then the following statements are equivalent: 1. A − 1 exists 2. A x = b has a unique solution for any b ∈ R n 3. A x = 0 only has the trivial solution, x = 0 4. The reduced row echelon form of A is I . Proof: ( 1 ) = ⇒ ( 2 ) = ⇒ ( 3 ) = ⇒ ( 4 ) = ⇒ ( 1 ) . 11

Elementary Matrices Matrix Inverse Determinants Cramer’s rule [ ∃ A − 1 ] = ⇒ [ ∃ ! x : A x = b , ∀ b ∈ R n ] • ( 1 ) = ⇒ ( 2 ) A − 1 A x = A − 1 b = ⇒ I x = A − 1 b = ⇒ x = A − 1 b hence x = A − 1 b is the only possible solution and it is a solution indeed: A ( A − 1 b ) = ( AA − 1 ) b = I b = b , ∀ b • ( 2 ) = [ ∃ ! x : A x = b , ∀ b ∈ R n ] = ⇒ ( 3 ) ⇒ [ A x = 0 = ⇒ x = 0 ] If A x = b has a unique solution for all b ∈ R n , then this is true for b = 0. The unique solution of A x = 0 must be the trivial solution, x = 0 12

Elementary Matrices Matrix Inverse Determinants Cramer’s rule • ( 3 ) = ⇒ ( 4 ) [ A x = 0 = ⇒ x = 0 ] = ⇒ [RREF of A is I ] then in the reduced row echelon form of A there are no non-leading (free) variables and there is a leading one in every column hence also a leading one in every row (because A is square and in RREF) hence it can only be the identity matrix ⇒ [ ∃ A − 1 ] • ( 4 ) = ⇒ ( 1 ) [RREF of A is I ] = ∃ sequence of row operations and elementary matrices E 1 , . . . , E r that reduce A to I ie, E r E r − 1 · · · E 1 A = I , E − 1 Each elementary matrix has an inverse hence multiplying repeatedly on the left by E − 1 r − 1 : r A = E − 1 · · · E − 1 r − 1 E − 1 I 1 r hence, A is a product of invertible matrices hence invertible. (Recall that ( AB ) − 1 = B − 1 A − 1 ) 13

Elementary Matrices Matrix Inverse Determinants Matrix Inverse via Row Operations Cramer’s rule We saw that: A = E − 1 · · · E − 1 r − 1 E − 1 I 1 r taking the inverse of both sides: A − 1 = ( E − 1 ) − 1 = E r · · · E 1 = E r · · · E 1 I · · · E − 1 r − 1 E − 1 1 r Hence: A − 1 = E r E r − 1 · · · E 1 I if E r E r − 1 E · · · E 1 A = I then Method: • Construct [ A | I ] • Use row operations to reduce this to [ I | B ] • If this is not possible then the matrix is not invertible • If it is possible then B = A − 1 14

Elementary Matrices Matrix Inverse Determinants Example Cramer’s rule       1 2 4 1 2 4 1 0 0 1 2 4 1 0 0 ii − i iii + i  → [ A | I ] = A = 1 3 6 1 3 6 0 1 0 − − → 0 1 2 − 1 1 0      − 1 0 1 − 1 0 1 0 0 1 0 2 5 1 0 1  1 2 4 1 0 0   1 2 0 − 11 8 − 4  i − 4 iii iii − 2 ii ii − 2 iii − − − → 0 1 2 − 1 1 0 − − − → 0 1 0 − 7 5 − 2     0 0 1 3 − 2 1 0 0 1 3 − 2 1  1 0 0 3 − 2 0  i − 2 ii − − − → 0 1 0 − 7 5 − 2   0 0 1 3 − 2 1   3 − 2 0 A − 1 = − 7 5 − 2   3 − 2 1 Verify by checking AA − 1 = I and A − 1 A = I . What would happen if the matrix is not invertible? 15

Elementary Matrices Matrix Inverse Determinants Verifying an Inverse Cramer’s rule Theorem If A and B are n × n matrices and AB = I , then A and B are each invertible matrices, and A = B − 1 and B = A − 1 . Proof: show that B x = 0 has unique solution x = 0, then B is invertible. AB = I B x = 0 = ⇒ A ( B x ) = A 0 = ⇒ ( AB ) x = 0 = ⇒ I x = 0 = ⇒ x = 0 So B − 1 exists. Hence: ⇒ ( AB ) B − 1 = IB − 1 = ⇒ A ( BB − 1 ) = B − 1 = ⇒ A = B − 1 AB = I = So A is the inverse of B , and therefore also invertible and A − 1 = ( B − 1 ) − 1 = B (Corollary: we do not need to verify both A − 1 A = I and AA − 1 = I , one sufficies) 16

Elementary Matrices Matrix Inverse Determinants Outline Cramer’s rule 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Cramer’s rule 17

Elementary Matrices Matrix Inverse Determinants Determinants Cramer’s rule • The determinant of a matrix A is a particular number associated with A , written | A | or det ( A ) , that tells whether the matrix A is invertible. • For the 2 × 2 case: � 1 b / a 1 / a 0 � � � a b 1 0 ( 1 / a ) R 1 [ A | I ] = − − − − − → c d 0 1 c d 0 1 � 1 � 1 � � b / a 1 / a 0 b / a 1 / a 0 R 2 − cR 1 aR 2 − − − − − → − − → 0 d − cb / a − c / a 1 0 ( ad − bc ) − c a Hence A − 1 exists if and only if ad − bc � = 0. • hence, for a 2 × 2 matrix the determinant is � �� � � � a b a b � � � � � = � = ad − bc � � � � c d c d � � 18