Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego
Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. Alexander Knop 2
Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding entries from row i of A and column j of B. In other words if (AB) i,j denotes (i, j)-entry of AB and A is m n matrix, then a i,1 b 1,j + + a i,n b n,j. Alexander Knop 2
Matrix Multiplication EXAMPLE Let us compute AB where A = [ ] 2 3 and B = 1 5 [ ] 4 3 6. 1 2 3 Alexander Knop 3
Matrix Multiplication EXAMPLE Let us compute AB where A = Ab 1 = [ 2 3 1 5 [ ] 2 3 and B = 1 5 [ ] 4 3 6. 1 2 3 ] [ ] [ ] 4 4 2 + 1 3 =, 1 4 1 + 1 ( 5) [ ] [ ] [ ] 2 3 3 3 2 + ( 2) 3 Ab 2 = =, 1 5 2 3 1 + ( 2) ( 5) [ ] [ ] 2 3 6 Ab 3 = = 1 5 3 [ 6 2 + 3 3 6 1 + 3 ( 5) ]. Alexander Knop 3
Matrix Multiplication 1 A(BC) = (AB)C 2 A(B + C) = AB + BC 3 (B + C)A = BA + CA 4 r(ab) = (ra)b = A(rB) 5 I m A = A = AI n Alexander Knop 4
Powers of Matrix DEFINITION If A is n n matrix, then A k denotes the product of k copies of A. Alexander Knop 5
The Transpose of Matrix DEFINITION If A is m n matrix, then the transpose of A, denoted by A T is a matrix whose columns are formed from the corresponding rows of A. Alexander Knop 6
The Transpose of Matrix DEFINITION If A is m n matrix, then the transpose of A, denoted by A T is a matrix whose columns are formed from the corresponding rows of A. EXAMPLE [ ] [ ] a b a c Let A =. Then A c d T = b d Alexander Knop 6
The Transpose of Matrix DEFINITION If A is m n matrix, then the transpose of A, denoted by A T is a matrix whose columns are formed from the corresponding rows of A. Let A and B denote matrices whose size are appropriate for the following operations. 1 (A T ) T = A 2 (A + B) T = A T + B T 3 for any scalar r, (ra) T = r(a T ) Alexander Knop 4 (AB) T = B T A T 6
The Inverse of a Matrix DEFINITION An n n matrix A is said to be invertible iff there is matrix C such that CA = I n AC = I n. Alexander Knop 7
The Inverse of a Matrix DEFINITION An n n matrix A is said to be invertible iff there is matrix C such that CA = I n AC = I n. NOTE Note that there is only one such C since if there is also B C such that BA = I n, then B = BI n = BAC = I n C = C, contradiction. We call such C as A 1. Alexander Knop 7
The Inverse of a Matrix [ ] a b Let A =. If ad bc 0, then A is invertible and c d A = 1 ad bc [ d ] b cc a and if ad bc = 0, then A is not invertible. If A is invertible n n matrix, then for each b in R n, the equation Ax = b has the unique solution x = A 1 b. Alexander Knop 8
The Inverse of a Matrix If A is invertible n n matrix, then for each b R n, the equation Ax = b has the unique solution x = A 1 b. Alexander Knop 9
The Inverse of a Matrix If A is invertible n n matrix, then for each b R n, the equation Ax = b has the unique solution x = A 1 b. PROOF. Take any b R n. Solution of the equation Ax = b exists since we can substitute A 1 b instead of x. Alexander Knop 9
The Inverse of a Matrix If A is invertible n n matrix, then for each b R n, the equation Ax = b has the unique solution x = A 1 b. PROOF. Take any b R n. Solution of the equation Ax = b exists since we can substitute A 1 b instead of x. In order to prove that solution u is unique let us note that A 1 Au = A 1 b, hence u = A 1 b. Alexander Knop 9
The Inverse of a Matrix 1 If A is invertible matrix, then A 1 is invertible and (A 1 ) 1 = A. 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. Alexander Knop 10
The Inverse of a Matrix 1 If A is invertible matrix, then A 1 is invertible and (A 1 ) 1 = A. 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. Alexander Knop 10
The Inverse of a Matrix 1 If A is invertible matrix, then A 1 is invertible and (A 1 ) 1 = A. 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. Alexander Knop 10
The Inverse of a Matrix 1 If A is invertible matrix, then A 1 is invertible and (A 1 ) 1 = A. 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. PROOF. The definition of invertible matrix states that AC = CA = I n but we can consider this equality as a statement that A is inverse of C. Alexander Knop 10
The Inverse of a Matrix 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. PROOF. Note that (AB)(B 1 A 1 ) = A(BB 1 )A 1 = AA 1 = I n. Alexander Knop 10
The Inverse of a Matrix 2 If A and B are invertible n n matrices, then so is AB and (AB) 1 = B 1 A 1. 3 If A is invertible matrix, then A T is invertible and (A T ) 1 = (A 1 ) T. PROOF. Note that A T (A 1 ) T = (A 1 A) T = I n. Alexander Knop 10
An Elementary Matrix DEFINITION An elementary matrix is one that is obtained by preforming single elementary row operation on an identity matrix. Alexander Knop 11
An Elementary Matrix DEFINITION An elementary matrix is one that is obtained by preforming single elementary row operation on an identity matrix. REMARK If an elementary row operation is performed on an m n matrix A, the resulting matrix can be written as EA, where the m n matrix E is created by performing the same row operation on I m. Alexander Knop 11
Elementary matrices and Invertibility Each elementary matrix is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I Alexander Knop 12
Elementary matrices and Invertibility Each elementary matrix is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I An n n matrix is invertible iff A is row equivalent to I n. Alexander Knop 12
Elementary matrices and Invertibility Each elementary matrix is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I An n n matrix is invertible iff A is row equivalent to I n. And in this case any sequence of operations that transform A to I n also transforms I n to A 1. Alexander Knop 12
Elementary matrices and Invertibility An n n matrix is invertible iff A is row equivalent to I n. Alexander Knop 13
Elementary matrices and Invertibility An n n matrix is invertible iff A is row equivalent to I n. And in this case any sequence of operations that transform A to I n also transforms I n to A 1. Alexander Knop 13
Elementary matrices and Invertibility An n n matrix is invertible iff A is row equivalent to I n. And in this case any sequence of operations that transform A to I n also transforms I n to A 1. PROOF. Let us assume that A is invertible. In this case Ax = b has solution for any b, hence A has pivot position on each row. As a result, since A is square, pivot position are on the main diagonal of A. Hence the reduced echelon form of A is I. Alexander Knop 13
Elementary matrices and Invertibility An n n matrix is invertible iff A is row equivalent to I n. And in this case any sequence of operations that transform A to I n also transforms I n to A 1. PROOF. Let us assume that A is invertible. In this case Ax = b has solution for any b, hence A has pivot position on each row. As a result, since A is square, pivot position are on the main diagonal of A. Hence the reduced echelon form of A is I. Now suppose conversely A I n. Hence there is a sequence E 1,, E p of elementary matrices such that E 1 E 2... E p A = I n. But it means, that A = (E p... E 1 ) 1. Alexander Knop 13