By J J Connor

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If an integer is followed by a smaller integer, the pair is said to form an inversion. The number of inversions for a set is defined as the sum of the inversions for each integer. As an illustration, we consider the set (3, 1, 4, 2). Working from left to right, the integer inversions are: Integer Inversions Total 3 (3, 1)(3, 2) 2 None (4,2) None 0 1 4 2 1 0 3 This set has three inversions. A permutation is classified as even (odd) if the total number of inversions for the set is an even (odd) integer.

For con- venience, consider the first r rows and columns to be linearly independent. 1—41. Find the rank of a by reducing it to an echelon matrix. a— 1122 2132 7797 4 2 1 PROBLEMS 1—42. 43 Show that c is at most of rank 1. a1 a2 C : When will r(c) 1—43. 0? Consider the product, r b11b12 a12 a11 b is of rank 1 and b11 Then, we can write 0. cblk) (j)2kj Show that the second, third,. , nth columns of c are multiples of the first column and therefore r(c) 1. When will r(c) = 0? Suppose r(a) = 1 and a11 0.

The complete problem can be efficiently handled by using the Gauss elimmation procedure (Refs. 9, 11, 13). REFERENCES 1. 2. 3. 4. 5. 6. 7, 8. R. , W. J. DUNCAN and A. R. cOLLAR: Elementary Matrices, cambridge University Press, London, 1963. THOMAS, G. : Calculus and Analytical Geometry. , 1953. : Matrix calculus, Interscience Publishers, New York, 1956. HOUN, F. , New York, 1958. , 1961. HOUSEHOLDER, A. , 1964. : Applied Linear Algebra, Prentice-Hall, New York, 1969. HIL DEBRAND, F. : Methods of Applied Mathematics, Prentice-Hall, New York, 1952.