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 A - 1 , the inverse of A, giving us x = A - 1 b. The other algorithm, called Cramer's rule, uses determinants to solve for x. These algorithms are now mostly of historical interest only, because LU decomposition followed by forward and back substitution is superior . Certain engineering and statistical applications do require the computation of a matrix's inverse, so it is still useful to know how to compute it.  |
