lect1118Inverses

# lect1118Inverses - MATH 17100 28446 Inverse of a Matrix...

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MATH 17100 28446 11/18/2009 Inverse of a Matrix Given a matrix A , suppose there exists matrix B such that = AB BA I . Evidently, this is possible, from conformability and compatibility conditions, only if both A and B are square and of the same order, nn × , say. We call B the inverse of A and denote it 1 = BA . Not every square matrix has an inverse. Those that do are said to be invertible or non-singular while those that don’t are said to be singular . Given a square matrix A , how can one tell whether 1 A exists, and how does one find it when it does exist? Thm. The × matrix A is invertible ( i.e. 1 A exists) if and only if det 0 A To find the inverse of a given non-singular square matrix, we can use (I) Row reduction Method 1. Form the augmented matrix [ |] AI 2. Obtain the row-reduced echelon form, which will be of the form [| ] IB if det 0 A 3. The inverse of A can be found as 1 = AB (II) Adjoint matrix Method 1. Obtain the cofactors of all elements of A and assemble the cofactor matrix 11 12 1 21 2 12 n n n n nn AA A A   =  C  2. The adjoint of A is defined as

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## This note was uploaded on 11/04/2010 for the course MATH 17100 taught by Professor Kitchens during the Spring '10 term at IUPUI.

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lect1118Inverses - MATH 17100 28446 Inverse of a Matrix...

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