This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: INVERTIBLE MATRICES JOS E MALAG ON-L OPEZ One good feature of square matrices is that we have the notion of inverse matrices. In particular, we have that the square matrices behave as numbers. A consequence of this is that we can recover a version of the cancellation property. Inverse Matrix Definition 1. Let A be a square matrix. We say that A is invertible if there is another square matrix B such that AB = BA = I In such case we say that B is the inverse of A . Notice that if A is invertible with inverse B , then B is also invertible with inverse A . We will denote the inverse matrix of A by A 1 . The inverse of a diagonal matrix is easy to obtain: Example 2. The inverse of the matrix A = 3 0 0 0 2 0 0 0 10 is given by A 1 = 1 / 3 1 / 2 1 / 10 In general, this is not the case. Example 3. The inverse of the matrix A = parenleftbigg 2 1 4 3 parenrightbigg is given by A 1 = 1 2 parenleftbigg 3 1 4 2 parenrightbigg 1 Remark 4 . (1) Only square matrices might have an inverse. (2) Not every square matrix will have an inverse. (3) Assume that A is an invertible matrix. If AC = AB then B = C . Similarly, if CA = BA then C = B . Theorem 5. If the inverse of a matrix exists, then the inverse is unique. Proof . Let A be an n n invertible matrix. Let B and B inverse matrices of A . Then B = BI n = B ( AB ) = ( BA ) B = I n B = B ....
View Full Document