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5 Basic concept of bideterminant in case of bimatrices We in this section proceed onto define the concept of determinant of a bimatrix and derive some of properties analogous to matrices which we bideterminant of a bimatrix. Let AB = A1 ∪ A2 be a square bimatrix. The bideterminant of a square bimatrix is an ordered pair (d1, d2) where d1 = |A1| and d2 = |A2|. |AB| = (d1, d2) where d1 and d2 are reals may be positive or negative or even zero. ( |A| denotes determinant of A). 1: Let ⎡ 3 0 0⎤ ⎡ 4 5⎤ ⎢ AB = ⎢ 2 1 1 ⎥⎥ ∪ ⎢ .
A 2nr ) (b) where ∑ ( ± ) a11υ a12 ν a1mw and ∑ (± ) a12υ a 22 ν a 2mw is the bideterminant of the subbimatrix formed from the first m rows and n columns of A1 and A2 respectively where AB = A1 ∪ A2. Thus the above expression (b) is the product of the bideterminant of the subbimatrix formed by crossing out the first m rows and columns. We have the correct sign since in the expression of |AB| the terms a111 a122 a1nn . and 2 2 2 a11 a 22 a nn has a plus sign. Next we shall consider the m × m subbimatrix formed from rows ( i11 , i12 ,… ,i1m ) and ( i12 , i 22 ,… ,i 2m ) and columns j11 , j12 ,… , j1m and j12 , j22 , , j2m .
Next we shall consider the m × m subbimatrix formed from rows ( i11 , i12 ,… ,i1m ) and ( i12 , i 22 ,… ,i 2m ) and columns j11 , j12 ,… , j1m and j12 , j22 , , j2m . Except for the sign the expansion of |AB| will contain the product of the bideterminant of this subbimatrix and the determinant of the subbimatrix formed by crossing out rows 1 1 1 1 1 1 2 2 2 ( i1 , i2 , ,im ) and ( i1 , i2 , ,im ) and columns j1 , j2 ,…, jm and j12 , j22 , , j2m . The sign of the product is determined by the method used in the expansion of bicofactors.