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In this section, we will study inverses of general square matrices. We begin with the formal definition. 1. Let A be a square matrix of size n × n. 7) where I = I n is the n × n identity matrix. The inverse is commonly denoted by X = A −1 . 7) in order to properly define an inverse to the matrix A. The first condition, X A = I , says that X is a left inverse, while the second, A X = I , requires that X also be a right inverse. Rectangular matrices might have either a left inverse or a right inverse, but, as we shall see, only square matrices have both, and so only square matrices 3/15/06 39 c 2006 Peter J.

10.   0 1 0 0 0 1 0, 0 1 0 0 1  There are six   0 1 0 0 1, 1 0 0 0 different 3 × 3 permutation     0 0 1 0 0 1 0 0, 1 0 0, 0 1 0 0 1 1 0 Thus, the first permutation matrix, which is the identity, does nothing. The fourth, fifth and sixth represent row interchanges. The second and third are non-elementary permutations, and can be realized by a pair of successive row interchanges. An elementary combinatorial argument proves that there are a total of n ! 22) different permutation matrices of size n × n.

11) A = 2 6 1, 5 0 0 1 1 4 2 The reader is urged to check this by directly multiplying the indicated matrices. In general, then, an m × m elementary matrix E of the first type will have all 1’s on the diagonal, one nonzero entry c in some off-diagonal position (i, j), with i = j, and all other entries equal to zero. If A is any m × n matrix, then the matrix product E A is equal to the matrix obtained from A by the elementary row operation adding c times row j to row i. ) To undo the operation of adding c times row j to row i, we must perform the inverse row operation that subtracts c (or, equivalently, adds − c) times row j from row i.

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