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Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing by Ronald Mak

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Chapter 11. Matrix Inversion, Determinants, and Condition Numbers

LU decomposition, as described in Chapter 10, not only allows us to solve a system of linear equations, but the algorithm makes it relatively simple to compute a square matrix's inverse and determinant. In this chapter, we'll also compute the matrix's condition number, a scalar value that indicates how well-conditioned (or ill-conditioned) is the system of linear equations represented by the matrix.

A matrix's inverse and its determinant are both used in two traditional algorithms for solving systems of equations. If we have the system Ax = b to solve for x, we can use matrix algebra and multiply both sides of the equation by A1, the inverse of A, giving us x = A1b. The other algorithm, ...

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