11.2 The Inverse We can use the LU decomposition algorithm to compute the inverse of square matrix A column by column. Each column j is the solution of the system Ax = i j , where i j is the j th column of the identity matrix I, since AA - 1 = I. This is actually an efficient way to compute the inverse. We need to decompose matrix A only once. Then, we can solve each system Ax = i j using forward substitution and back substitution. For example, if A is a 3 x 3 matrix, then the second column of A - 1 is the solution x of the system
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