MAT1341-L13-MatrixInverse0

MAT1341-L13-MatrixInverse0 - INVERTIBLE MATRICES JOS ´ E...

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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 ′ ....
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MAT1341-L13-MatrixInverse0 - INVERTIBLE MATRICES JOS ´ E...

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