Determinant problem for block-matrices             Torsten Sillke, 1995-12-28
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Let A, B, C, D be n*n square matrices.

 | A  B |
 | C  D |  =  | A*D - C*B |

if  A*C = C*A   (**).

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Typically the proof for this result is given as follows:

Case I)  A is invertible.

   Let A' = Inverse(A).

   (   I    0 )     ( A  B )     ( A       B    )
   ( -C*A'  I )  *  ( C  D )  =  ( 0  -C*A'*B+D )

   | A  B |     | A       B    |
   | C  D |  =  | 0  -C*A'*B+D |  =  |A| * | -C*A'*B+D |

             =  | A*D - A*C*A'B | = | A*D - C*B |

Case II)  otherwise

   Perturbation method:
   Replace A by A + epsilon*I.
   Now (**) is valid for the perturbated A too.
   So we are in case I again. Now take the limit epsilon -> 0.

Problem:
   Can one avoid this analytic method?
   Is there a pure algebraic derivation?
   In other words, does the equation hold for all commutative rings?


Solution:
---------
   Torsten Sillke, William P. Wardlaw;
   Q873 Quickie on Block Determinants,
   Mathematical Magazin, 70:5 (Dec. 1997) 382


References:
----------- 
- I. Kovacs, D. S. Silver, S. G. Williams,
   Determinants of Commuting-Block Matrices,
   American Mathematical Monthly 106 (Dec. 1999) 950-952

- M. Marcus,
   Two determinant condensation formulas
   Linear and Multilinear Algebra 22 (1987), no. 1, 95--102.

- L. Mirsky, 
   Linear Algebra, 1955, Clarendon Pr. Oxford 
   Chapter III Problem 36, p110 

- AMM 82 (Nov. 1975) 942, Adv. Problem 6057, Determinant of Matrices 

- AMM 84 (June-July 1977) 495-496, Adv. Prob. 6057 Sol. Determinant of Matrices

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 89g:15009 15A15 
 Marcus, Marvin(1-UCSB-C) 
 Two determinant condensation formulas. (English) 
 Linear and Multilinear Algebra 22 (1987), no. 1, 95--102. 

 Let $A,B,C,D$ be square matrices of order $n$ with entries from an 
 integral domain $R$ with 1 and of characteristic not 2. If $CD\sp 
 T=DC\sp T$ and $D$ is nonsingular then $\det \left[\smallmatrix A&B\ 
 C&D\endsmallmatrix\right]= \det(AD\sp T-BC\sp T)$. Similarly if $CD\sp 
 T=-DC\sp T$ and $D$ is nonsingular then $\det\left[\smallmatrix A&B\ 
 C&D\endsmallmatrix \right]=\det(AD\sp T+BC\sp T)$. The first result 
 remains true when $D$ is singular but the second one fails. The 
 case in which $R$ is the real or complex field and $D$ is singular 
 have been treated with continuity arguments. The author proves by 
 matrix methods for arbitrary $R$ that the first equality holds and 
 the second holds if and only if $n-\roman{rank}(D)$ is even. 

 Reviewed by M. Pearl 


 87c:15001 15-01 15A24 
 Wardlaw, William P.(1-USNA) 
 A transfer device for matrix theorems. (English) 
 Math. Mag. 59 (1986), no. 1, 30--33. 

 Let $F$ be a formula---such as the Cayley-Hamilton theorem or the product 
 formula for determinants---involving matrices over a commutative ring. 
 The author shows how to establish several such formulas $F$ over an 
 arbitrary commutative ring once one knows them in the special case of 
 matrices of real numbers, perhaps invertible or with distinct eigenvalues. 
 The proofs make use of the fact that the real numbers have 
 infinite transcendence degree over the rational numbers. 

 The motivation is to make graduate algebra classes more interesting by 
 enabling continuity to be used in the proof of theorems about arbitrary 
 commutative rings. (For another interesting idea of this type, see a paper 
 by W. Watkins [Amer. Math. Monthly 82 (1975), 364--368; MR 51 #554].) 

 Reviewed by Lawrence S. Levy