DM554 Linear and Integer Programming Lecture 5 Matrix Inverse and Determinants Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark
Elementary Matrices Matrix Inverse Determinants Outline More on Inverse 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Matrix Inverse and Cramer’s rule 2
Elementary Matrices Matrix Inverse Determinants Outline More on Inverse 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Matrix Inverse and Cramer’s rule 3
Elementary Matrices Matrix Inverse Determinants Elementary matrix More on Inverse 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 4
Elementary Matrices Matrix Inverse Determinants More on Inverse 1 2 4 1 2 4 ii − i B = 1 3 6 − − → 0 1 2 − 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 5
Elementary Matrices Matrix Inverse Determinants Matrix Inverse More on Inverse 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 E − 1 = 1 ( E 1 B ) = 1 1 0 0 1 2 1 3 6 0 0 1 − 1 0 1 − 1 0 1 6
Elementary Matrices Matrix Inverse Determinants Row equivalence More on Inverse To be an equivalence relation a relation must satisfy three properties: • reflexive: A ∼ B • symmetric: A ∼ B = ⇒ B ∼ A • transitive: A ∼ B and B ∼ C = ⇒ A ∼ C 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 . Theorem Every matrix is row equivalent to a matrix in reduced row echelon form 7
Elementary Matrices Matrix Inverse Determinants Invertible Matrices More on Inverse 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 ) . • ( 1 ) = ⇒ ( 2 ) A − 1 A x = A − 1 b = ⇒ I x = A − 1 b = ⇒ x = A − 1 b hence x = A − 1 b is a solution and it is unique, indeed: A ( A − 1 b ) = ( AA − 1 ) b = I b = b , ∀ b • ( 2 ) = ⇒ ( 3 ) 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 8
Elementary Matrices Matrix Inverse Determinants More on Inverse • ( 3 ) = ⇒ ( 4 ) 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 • ( 4 ) = ⇒ ( 1 ) ∃ 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 Each elementary matrix has an inverse hence multiplying repeatedly on , E − 1 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 B − 1 A − 1 = ( AB ) − 1 ) 9
Elementary Matrices Matrix Inverse Determinants Outline More on Inverse 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Matrix Inverse and Cramer’s rule 10
Elementary Matrices Matrix Inverse Determinants Matrix Inverse via Row Operations More on Inverse 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 r 1 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 11
Elementary Matrices Matrix Inverse Determinants Example More on Inverse 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? 12
Elementary Matrices Matrix Inverse Determinants Verifying an Inverse More on Inverse 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 for the previous theorem. 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 13
Elementary Matrices Matrix Inverse Determinants Outline More on Inverse 1. Elementary Matrices 2. Matrix Inverse 3. Determinants 4. Matrix Inverse and Cramer’s rule 14
Elementary Matrices Matrix Inverse Determinants Determinants More on Inverse • 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 � � 15
• The extension to n × n matrices is done recursively Definition (Minor) For an n × n matrix the ( i , j ) minor of A , denoted by M ij , is the determinant of the ( n − 1 ) × ( n − 1 ) matrix obtained by removing the i th row and the j th column of A . Definition (Cofactor) The ( i , j ) cofactor of a matrix A is C ij = ( − 1 ) i + j M ij Definition (Cofactor Expansion of | A | by row one) The determinant of an n × n matrix is given by � � a 11 a 12 · · · a 1 n � � � � a 21 a 22 · · · a 2 n � � | A | = = a 11 C 11 + a 12 C 12 + · · · + a 1 n C 1 n . . . � ... � . . . � � . . . � � � � a n 1 a n 2 · · · a nn � �
Elementary Matrices Matrix Inverse Determinants More on Inverse Example 1 2 3 | A | = 1 C 11 + 2 C 12 + 3 C 13 A = 4 1 1 � � � � � � 1 1 4 1 4 1 � � � � � � − 1 3 0 = 1 � − 2 � + 3 � � � � � � 3 0 − 1 0 − 1 3 � � � � = 1 ( − 3 ) − 2 ( 1 ) + 3 ( 13 ) = 34 Theorem If A is an n × n matrix, then the determinant of A can be computed by multiplying the entries of any row (or column) by their cofactors and summing the resulting products: | A | = a i 1 C i 1 + a i 2 C i 2 + · · · + a in C in (cofactor expansion by row i) | A | = a 1 j C 1 j + a 2 j C 2 j + · · · + a nj C nj (cofactor expansion by column j) 17
Elementary Matrices Matrix Inverse Determinants More on Inverse A mnemonic rule for the 3 × 3 matrix determinant: the rule of Sarrus | A | = + a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 11 a 23 a 32 − a 12 a 21 a 33 − a 13 a 22 a 31 Verify the rule: • from the conditions of existence of an inverse • as a consequence of the general recursive rule for the determinants 18
Elementary Matrices Matrix Inverse Determinants Geometric interpretation More on Inverse 2 × 2 The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram’s sides. 3 × 3 The volume of this parallelepiped is the absolute value of the determinant of the matrix formed by the rows constructed from the vectors r 1, r 2, and r 3. 19
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