Chapter 3 The Inverse

1. Chapter 3The Inverse

2. 3.1 Introduction Definition 1: The inverse of an nn matrix A is an nn matrix B having the property that AB = BA = I B is called the inverse of A and is usually denoted by A-1. If a square matrix has an inverse, it is said to be invertible or nonsingular. If it doesn’t possess an inverse, it is said to be singular. Example: The inverse of is because

3. 3.1 Introduction: elementary matrices • An elementary matrix E is a square matrix that generates an elementary row operation on a matrix A (which need not to be square) under the multiplication EA. • We will need elementary matrices for finding inverses. • To construct an elementary matrix that interchanges the ith row and the jth row, begin with an identity matrix. First interchange the unity element in the i-i position with the zero in j-i position, and then interchange the unity element in the j-j position with the zero in i-j position. • To construct an elementary matrix that multiplies the ith row of a matrix by the nonzero scalar k, replace the unity element in the i-ipoistion of the identity matrix of appropriate order with the scalar k. • To construct an elementary matrix that adds to the jth row ofa matrix k times the ith row, replace the zero element in the j-i position of the identity matrix with the scalar k. • Examples on the board.

4. 3.1 Introduction: inverse of a matrix in block diagonal form If A can be partitioned into the block-diagonal form, Then A is invertible if and only if each of the diagonal blocks A1, …, Anis invertible and Examples on the board.

5. 3.2 Calculating Inverses • Theorem 1: A square matrix has an inverse if and only if reduction to row-reduced form by elementary row operations results in a matrix having all unity elements on the main diagonal. • Theorem 2: An nn matrix has an inverse if and only if it has rank n. • Theorem 1 suggests a procedure for finding the inverse. The idea is to • first transform the matrix to a row-reduced form • then keep applying elementary row operation (E3) to reduce it further to the identity matrix. • Recall from Section 3.1 that these transformations can be accomplished by multiplying the matrix with elementary matrices: • Ek Ek-1 … E3 E2 E1 A = I • But then A-1 = Ek Ek-1 … E3 E2 E1 • Thus, to calculate the inverse we need to keep a record of the elementary matrices. This is accomplished by applying the same elementary operations to both A and an identity matrix since (Ek Ek-1 … E3 E2 E1)I = Ek Ek-1 … E3 E2 E1 = A-1

6. Example 1: Example 2: A Non-Invertible Matrix

7. 3.3 Simultaneous Equations A linear system can be solved using inverses. Suppose we have a linear system A x = B where A is invertible. Then A-1 A x = A-1 B I x = A-1 B x = A-1 B Theorem: If A is invertible, then A x = B has one and only one solution. Examples on the board.

8. 3.4 Properties of the Inverse • (A-1)-1 = A • (AB)-1 = B-1 A-1 • (AT)-1 = (A-1)T • (λA)-1 = (1/ λ) A-1 • The inverse of a matrix is unique. • The inverse of a nonsingular symmetric matrix is symmetric. • The inverse of a nonsingular upper or lower triangular matrix is again upper or lower triangular matrix respectively. • If A is nonsingular then we define A-n = (A-1)n