Basing on the relation (1.4), we can use another notation for the set of magic rectangles with fixed row sum and column sum. Namely, simultaneously with the notation MR(m, n; r, c) we will use a notation MR[m, n : g] which implies that r = ng and c = mg.
   Obviously, for m = n a magic rectangle becomes a semi-magic square.
   2. Let us define a class of block matrices composed of magic rectangles.
   Definition 2.1 A matrix A = [a_ij] О R^m×n represented in the block form

with submatrices  A_ij О MR(m_i, n_j; r_ij, c_ij)  (or  A_ij О MR[m_i, n_j : g_ij] , in another notation), where  i = 1,2,...,p ,  j = 1,2,...,q and = m, = n, will be referred to as block magic rectangle.

   Insert diagonal matrices

we will denote the set of block magic rectangles partitioned into blocks correspondingly to (2.2) with row sums and column sums defined by the matrices (2.3).
   In accordance with relation (1.4) and notation accepted in the previous section, we have

rij = gijnj ,    cij = gijmi

(2.5)

for  i = 1,2,...,p and  j = 1,2,...,q . Consider a matrix

G = й к к к к л g11 g12 ... g1q g21 g22 ... g2q ... ... ... ... gp1 gp2 ... gpq щ ъ ъ ъ ъ ы  .

(2.6)

Then the relations (2.5) can be written in the matrix form:

R = GN ,    C = MG .

(2.7)

   Therefore, we will also use another notation for the set of block magic rectangles (2.4), that is

BMR[M,N : G] .

(2.8)

   The properties (1.3) for the blocks of matrix (2.1) look as

Aijenj = rijemi ,    AijTemi = cijenj ,

(2.9)

where  i = 1,2,...,p and  j = 1,2,...,q . Let us define block matrices

Emp = й к к к к к л em1 0 ... 0 0 em2 ... 0 ... ... ... ... 0 0 ... emp щ ъ ъ ъ ъ ъ ы О Rm×p , Enq = й к к к к к л en1 0 ... 0 0 en2 ... 0 ... ... ... ... 0 0 ... enq щ ъ ъ ъ ъ ъ ы О Rn×q

with blocks of the size  m_i × 1 ,  i = 1,2,...,p (in the matrix  E_mp) and  n_j × 1 ,  j = 1,2,...,q (in the matrix  E_nq). By a straightforward verification it can be easily shown that relations (2.9) are equivalent to the following ones:

AEnq = EmpR ,    ATEmp = EnqCT .

   3. Remind, that the Moore-Penrose inverse  A⁺ О R^n×m of a matrix  A О R^m×n is uniquely determined by the properties

a) AA+A = A , b) A+AA+ = A+ , c) (A+A)T = A+A , d) (AA+)T = AA+

(see [3] , for instance).
   The main result of the paper is formulated as follows.
   Theorem 3.1  If  A О BMR(M, N ; R , C) then  A⁺ О BMR(N, M ; ) , where

= N-1/2(M1/2RN-1/2)+M1/2 ,

(3.1)

= N1/2(M-1/2CN1/2)+M-1/2 .

(3.2)

   Let us give an equivalent formulation of Theorem 3.1 , connected with the notation of type (2.8) for a set of block magic rectangles.
   Theorem 3.1A If  A О BMR[M, N : G] then  A⁺ О BMR[N, M : ] , where

= N-1/2(M1/2GN1/2)+M-1/2 .

(3.3)

   Example 3.1. Consider a block magic rectangle

A = й к к к к к к л 1 0 5 1 1 0 3 4 -1 1 1 0 1 0 2 4 2 2 1 2 0 1 5 2 1 0 2 3 3 2 1 2 0 4 2 2 щ ъ ъ ъ ъ ъ ъ ы О R6×6          (p = 2 , q = 3) .

According to (2.2),(2.3),(2.5),(2.6), we have

M = й к л 2 0 0 4 щ ъ ы  ,   N = й к к л 3 0 0 0 2 0 0 0 1 щ ъ ъ ы

and

R = й к л 6 2 0 3 6 2 щ ъ ы  ,   C = й к л 4 2 0 4 12 8 щ ъ ы  ,   G = й к л 2 1 0 1 3 2 щ ъ ы  .

   The Moore-Penrose inverse of the matrix  A calculated using MATLAB is

A+ = й к к к к к к л -0.2411 0.4256 0.3212 -0.1788 0.4879 -0.6788 0.2589 -0.0744 -0.2788 0.1212 -0.4121 0.5212 0.2589 -0.0744 -0.0788 0.0212 -0.1121 0.1212 -0.0267 -0.0267 0.0864 -0.1136 -0.0136 0.1864 -0.0267 -0.0267 -0.0136 0.1864 0.0864 -0.1136 -0.0583 -0.0583 0.0340 0.0340 0.0340 0.0340 щ ъ ъ ъ ъ ъ ъ ы  .

For this matrix

= й к к к л 0.1845 -0.0485 -0.0534 0.1456 -0.1165 0.1359 щ ъ ъ ъ ы  ,    = й к к к л 0.2767 -0.0364 -0.0534 0.0728 -0.0583 0.0340 щ ъ ъ ъ ы  ,

= й к к к л 0.0922 -0.0121 -0.0267 0.0364 -0.0583 0.0340 щ ъ ъ ъ ы  .

   It is of interest to examine some particular cases of block magic rectangles for which the general formula, describing the Moore-Penrose inverse, is considerably simplified.
   Case 1. Suppose that in block representation (2.1)  p = q = 1 , i.e. let  A О MR(m, n ; r, c) . Then, by formulae (3.1) and (3.2), we get

= n-1/2(m1/2rn-1/2)+m1/2 = n-1/2n1/2r+m-1/2m1/2 = r+ , = n1/2(m-1/2c n1/2)+m-1/2 = n1/2n-1/2c+m1/2m-1/2 = c+ .

   Recall that for a scalar  a , which can be considered as  1 × 1 matrix,  a⁺ = 1/a , if  a № 0 , and  a⁺ = 0 , if  a = 0 . So, we arrive at the following statement.
   Theorem 3.2 If  A О MR(m, n ; r , c) then  A⁺ О MR(n, m ; r⁺ , c⁺) .
   Note, that the last proposition has been obtained recently in [4].
   We can also give an equivalent formulation of Theorem 3.2 . Let  A О MR[m, n : g] . According to (3.3), we find

=n-1/2(m1/2gn1/2)+m-1/2 = n-1/2n-1/2g+m-1/2m-1/2 = 1 mn  g+ .

   Thus, the result can be stated as follows.
   Theorem 3.2A If  A О MR[m, n : g] then  A⁺ О MR[n, m : (mn)^-1 g⁺] .
   Case 2. Let in block form (2.1)  m₁ = m₂ = ј =  m_p є k and  n₁ = n₂ = ј =  n_q є l . It is clear, that  k = m/p and  l = n/q . Thereby, according to (2.2), we have  M = kI_p and  N = lI_q where  I_p and  I_q are identity matrices of  pth and  qth order, respectively. In this case our formulae (3.1) and (3.2) become extremely simple, i.e.
Thus, we get
   Theorem 3.3  If  A О BMR(kI_p, lI_q ; R , C) then  A⁺ О BMR(lI_q, kI_p ; R⁺ , C⁺) .
   Next, from (3.3) we obtain
   Consequently, the result may be formulated as follows.
   Theorem 3.3A  If  A О BMR[kI_p, lI_q : G] then  A⁺ О BMR[lI_q, kI_p : (kl)^-1 G⁺] .

   Yerevan State University