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to denote dierent unknown coecients as opposed to

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Unformatted text preview: → −−−−−−−− I2 A−1 In other words, the process of finding A−1 for a matrix A can be viewed as performing a series of row operations which transform A into the identity matrix of the same dimension. We can view this process as follows. In trying to find A−1 , we are trying to ‘undo’ multiplication by the matrix A. The identity matrix in the super-sized augmented matrix [A|I ] keeps a running memory of all of the moves required to ‘undo’ A. This results in exactly what we want, A−1 . We are now ready 8.4 Systems of Linear Equations: Matrix Inverses 497 to formalize and generalize the foregoing discussion. We begin with the formal definition of an invertible matrix. Definition 8.11. An n × n matrix A is said to be invertible if there exists a matrix A−1 , read ‘A inverse’, such that A−1 A = AA−1 = In . Note that, according to our definition, invertible matrices are square, and as such, the conditions in Definition 8.11 force the matrix A...
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