2005/7Inverse matrices-1 Inverse and Elementary Matrices

2005/7Inverse Matrices-2 反矩陣 (inverse matrix) If there is a matrixsuch that Note : If a matrix having no inverse matrix is called a noninvertible or singular matrix. Let then (1) A is called an invertible or nonsingular matrix (2) B is the inverse matrix of A

2005/7Inverse Matrices-3 Theorem ： If B and C are both the inverse of A, then B = C Pf: Since B = C, the inverse matrix of a matrix is unique. Note: (1) The inverse matrix of A is denoted by (2)

2005/7Inverse Matrices-4 Use the Gaussian-Jordon elimination to find the inverse matrix Ex ： Find the inverse of Sol: 12

2005/7Inverse Matrices Hence

2005/7Inverse Matrices-6 Note ： If A can’t using row operations to be translated into identity matrix I, then A is a singular matrix.

2005/7Inverse Matrices-7 Ex ： Find the inverse matrix of Sol:

2005/7Inverse Matrices-8

2005/7Inverse Matrices-9 Thus

2005/7Inverse Matrices-10 The power of a square matrix

2005/7Inverse Matrices-11 Theorem ： If A is invertible, then the following properties hold:

2005/7Inverse Matrices-12 Theorem ： If A and B are both invertible with the size n  n, then AB is invertible and Pf: Note: Thus AB is invertible and the inverse matrix of AB is (BA)  1.

2005/7Inverse Matrices-13 Theorem ： Cancellation laws If C is invertible, then the following properties hold: (1) If AC=BC, then A=B (2) If CA=CB, then A=B Pf : Note : If C is noninvertible, then the cancellation laws do not hold. Since C is invertible, C  1 exists.

2005/7Inverse Matrices-14 Theorem ： If A is invertible, then Ax = b has a unique solution. Pf : ( A is nonsingular) If x 1 and x 2 are two solutions of Ax = b, then Ax 1 = b = Ax 2. By the cancellation law, x 1 = x 2, the solution is unique. Note :

2005/7Inverse Matrices-15 Three different row elementary matrices row elementary matrix( 列基本矩陣 ) An n  n matrix is called a row elementary matrix if it can be attained from identity I by doing only one row elementary operation elementary matrix Elementary operation identity matrix

2005/7Inverse Matrices-16 Ex ： (a) (b) (c) (d) (e)(f)

2005/7Inverse Matrices-17 Ex ：求一序列的基本矩陣以將下列矩陣化簡成列梯形形式 Sol :

2005/7Inverse Matrices-18 = B

2005/7Inverse Matrices-19 If there are finite row elementary matrices, E 1, E 2, …, E k such that, then B is row-equivalent to A. 列等價 (row-equivalent)

2005/7Inverse Matrices-20 Theorem ： Elementary matrix is invertible. If E is an elementary matrix, then E  1 exists and it is also an elementary matrix. Note:

2005/7Inverse Matrices-21 Ex: elementary matrix inverse matrix

2005/7Inverse Matrices-22 Theorem: A square matrix A is invertible if and only if it can be represented as a product of a sequence of elementary matrices. Pf : (1)Assume that A can be written as a product of a sequence of elementary matrices. Every elementary matrix is invertible. The product of invertible matrices is invertible. Thus A is invertible. (2) If A is invertible then Ax = 0 has only trivial solution. Thus A can be written as a product of elementary matrices.

2005/7Inverse Matrices-23 Ex ： Find a sequence of elementary matrices such that their product is the given matrix Sol :

2005/7Inverse Matrices-24 Theorem: If A is an n  n matrix, then the following statements are equivalent: (1) A is invertible. (2) For any n  1 matrix b, Ax = b has only one solution. (3) Ax = 0 has only trivial solution. (4) A is (row) equivalent to I n. (5) A can be represented as a product of elementary matrices.

2005/7Inverse Matrices-25 L is a lower triangular matrix U is an upper triangular matrix Represent an n  n matrix A as a product of a lower triangular matrix L and an upper triangular matrix U. LU-factorization (LU- 分解 ) Note: If A is LU-factorizatiable than we can use only one elementary operation, r ij (k), to translate A into LU.

2005/7Inverse Matrices-26 Ex: Do the LU-factorization of A. (b)(a) Sol: (a)

2005/7Inverse Matrices-27 (b)

2005/7Inverse Matrices-28 Use LU-factorization to solve Ax=b. Two steps: (1) Let y=Ux. Solve Ly=b, find y. (2) Solve Ux=y, then we can get x.

2005/7Inverse Matrices-29 Ex ： Use LU-factorization to solve the given linear system. Sol: (1) Let y=Ux. Solve Ly=b, find y.

2005/7Inverse Matrices-30 Thus the solution of the given system is (2) Solve Ux=y, then we can get x.