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1. The Inverse of a Nonsingular Matrix It is well known that every nonsingular matrix A has a unique inverse,?1 denoted by A, such that?1?1 AA = A A =I, (1) where I is the identity matrix. Of the numerous properties of the inverse matrix, we mention a few. Thus,?1?1 (A ) = A, T?1?1 T (A ) =(A ),??1?1? (A ) =(A ),?1?1?1 (AB) = B A, T? where A and A, respectively, denote the transpose and conjugate tra- pose of A. It will be recalled that a real or complex number? is called an eigenvalue of a square matrix A, and a nonzero vector x is called an eigenvector of A corresponding to?,if Ax =?x.?1 Another property of the inverse A is that its eigenvalues are the recip- cals of those of A. 2. Generalized Inverses of Matrices A matrix has an inverse only if it is square, and even then only if it is nonsingular or, in other words, if its columns (or rows) are linearly in- pendent. In recent years needs have been felt in numerous areas of applied mathematics for some kind of partial inverse of a matrix that is singular or even rectangular.