By George A. F. Seber

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The row rank is dimC(A'). If A is m x n of rank m (respectively n ) , then A is said to have full row (respectively column) rank. An n x n matrix A is said to be nonsingular if rankA = n. 4, an associated vector space of C(A) is the null space N ( A ) , and its dimension is called the nullity. 1. rank A' = rank A = r so that the row rank equals the column rank. 2. Let A be an m x n matrix of rank r ( r 5 min{m,n}). (a) A has r linearly independent columns and T linearly independent rows. (b) There exists an r x r nonzero principal minor.

5), we can take a left inverse of A and then a left inverse of B to get x = 0 so that the columns of AB are linearly independent. 16. Graybill [1983: 891. 17. Harville [200l: 27, exercise I], Marsaglia and Styan [1974a: theorem 21, and Rao and Rao [1998: 1331. 18. Harville [1997: 3961 and Marsaglia and Styan [1974a: theorem 71. 19. Isotalo et al. [2005b: 171. 20. F. (a) If C has full column rank and R has full row rank, then using left and right inverses, respectively, we have that C A = CB implies A = B and PR = QR implies P = Q.

I) rank(A When c = 0, M = 0 and r a n k M = 0 = c so that rank(A (t) + B) = rank (3 + (iv) rank = rankA r a n k B if and only if d = 0. When d = 0. M = 0 and r a n k M = 0 = d so that rank(A (v) rank(A + B) = rank(A, B). + B) = r a n k A + r a n k B if and only if c = d = 0. 24. 12) that rank(AB - In)5 rank(A - - I, = (A - 1,)B + + I n ) rank(B - In). 2. Suppose that A = C,=lA,, where each matrix is m x n. We say rank A,. 25. Let A and B be nonnull m x n matrices over F with respective ranks s.