Note that inversion reverses the order of compo- sition. If S and T are isomorphisms, then so is S ◦ T , and (S ◦ T )−1 = (T −1) ◦ (S−1).

−1  T VW- T Matrix inversion @I@ 6 @@ S ◦ T Math 130 Linear Algebra @@ S S−1 D Joyce, Fall 2015 @@ @@ (S ◦ T )−1 @@ @R ? Recall that a linear transformation T is an iso- @ X morphism when it has an inverse, that is, there’s another linear transformation S such that T ◦ S We can translate that into the language of ma- and S ◦ T are both the identity transformation. trices as the following theorem. As we’ve noted before, two finite dimensional vector spaces can be isomorphic if and only if they Theorem 3. If A and B are both invertible, then have the same dimension. We’ll denote that dimen- their product is, too, and (AB)−1 = B−1A−1. sion by n. Matrices encode linear transformations on finite Inverses of 2 × 2 matrices. You can easily find dimensional vector spaces, so we can translate that the inverse of a 2 × 2 matrix. Consider a generic statement to define invertibility of matrices. 2 × 2 matrix a b A = Definition 1. We say that two square n × n ma- c d trices A and B are inverses of each other if It’s inverse is the matrix AB = BA = I  d/∆ −b/∆ A−1 = −c/∆ a/∆ and in that case we say that B is an inverse of A and that A is an inverse of B. If a matrix has no where ∆ is the determinant of A, namely inverse, it is said to be singular, but if it does have an inverse, it is said to be invertible or nonsingular. ∆ = ad − bc,

With that definition, a matrix is invertible if and provided ∆ is not 0. In words, to find the inverse only if it represents an invertible linear transforma- of a 2 × 2 matrix, (1) exchange the entries on the tion F n →' F n, in which case its inverse represents major diagonal, (2) negate the entries on the mi- the inverse linear transformation. nor diagonal, and (3) divide all four entries by the A linear transformation T can have at most one determinant. −1 inverse, which we denote T −1, and (T −1)−1 = T . It’s easy to verify that A actually is the inverse Therefore, we can conclude the following theorem. of A, just multiply them together to get the identity matrix I. Theorem 2. A matrix A can have at most one inverse. The inverse of an invertible matrix is A method for finding inverse matrices. Next denoted A−1. Also, when a matrix is invertible, we’ll look at a different method to determine if an so is its inverse, and its inverse’s inverse is itself, n×n square matrix A is invertible, and if it is what (A−1)−1 = A. it’s inverse is.

1 The method is this. First, adjoin the identity Matlab can compute inverses or tell you if matrix to its right to get an n × 2n matrix [A|I]. they’re singular. Next, convert that matrix to reduced echelon form. If the result looks like [I|B], then B is the desired >> A = [1 2; 3 4] inverse A−1. But if the square matrix in the left A = half of the reduced echelon form is not the identity, 1 2 then A has no inverse. 3 4 We’ll verify that this method works later. >> B = inv(A) Example 4. Let’s illustrate the method with a 3× B = 3 example. Let A be the matrix -2.0000 1.0000 1.5000 -0.5000 3 −2 4 A = 1 0 2 >> A*B 0 1 0 ans = Form the 3 × 6 matrix [A|I], and row reduce it. 1.0000 0 I’ll use the symbol ∼ when a row-operation is ap- 0.0000 1.0000 plied. Here are the steps. >> C = [1 2; 3 6]  3 −2 4 1 0 0  C = [A|I] =  1 0 2 0 1 0  1 2 0 1 0 0 0 1 3 6  1 0 2 0 1 0  ∼  0 1 0 0 0 1  >> D = inv(C) 3 −2 4 1 0 0 Warning: Matrix is singular to working precision. D =  1 0 2 0 1 0  Inf Inf ∼ 0 1 0 0 0 1   Inf Inf 0 −2 −2 1 −3 0  1 0 2 0 1 0  Math 130 Home Page at ∼  0 1 0 0 0 1  0 0 −2 1 −3 2 http://math.clarku.edu/~ma130/  1 0 0 1 −2 2  −1 ∼  0 1 0 0 0 1  = [I|A ] 0 0 1 −1/2 3/2 −1

This row-reduction to reduced echelon form suc- ceeded in turning the left half of the matrix into the identity matrix. When that happens, the right half of the matrix will be the inverse matrix A−1. Therefore, the inverse matrix is

 1 −2 2  −1 A =  0 0 1  . −1/2 3/2 −1

2