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Determinants of Block Matrices Determinants of Blo ck Matrices John R. Silvester 1 Intro duction ! ! e f a b . Their sum and N = Let us rst consider the 2 2 matrices M = g h c d and pro duct are given by ! ! a + e b + f ae + bg af + bh M + N = and MN = : c + g d + h ce + dg cf + dh Here the entries a; b; c; d; e; f ; g ; h can come from a eld, such as the real numb ers, or more n n generally from a ring, commutative or not. Indeed, if F is a eld, then the set R = F of all n n matrices over F forms a ring non-commutative if n 2, b ecause its elements can b e added, subtracted and multiplied, and all the ring axioms asso ciativity, distributivity, etc. hold. If a;b;:::;h are taken from this ring R , then M; N can be 2 2 2n 2n thought of either as memb ers of R 2 2 matrices over R or as memb ers of F . It is well-known fact, which we leave the reader to investigate, that whether we compute with these matrices as 2n 2n matrices, or as 2 2 \blo ck" matrices where the blo cks a;b;::: are n n matrices, i.e., elements of R makes no di erence as far as addition, subtraction and multiplication of matrices is concerned. See for example [2], p. 4, or 2 2 2n 2n [6], pp. 100{106. In symb ols, the rings R and F can be treated as b eing identical: 2 2 2n 2n 2 n n 2 2n 2n R = F , or F = F . More generally, we can partition any mn mn matrix m n n m mn mn as an m m matrix of n n blo cks: F = F . The main p oint of this article is to lo ok at determinants of partitioned or blo ck matrices. If a; b; c; d lie in a ring R , then provided that R is commutative there is a determinant for M, which we shall write as det , thus: det M = ad bc, which of R R course lies in R . If R is not commutative, then the elements ad bc, ad cb, da bc, da cb may not b e the same, and we do not then know which of them if any mightbe n n a suitable candidate for det M. This is exactly the situation if R = F , where F is a R eld or a commutative ring and n 2; so to avoid the diculty we take R to be, not n n n n the whole of the matrix ring F , but some commutative subring R F . The usual 2 2 m m theory of determinants then works quite happily in R , or more generally in R , and m m n n for M 2 R we can work out det M, which will b e an elementofR . But R F , so R det M is actually a matrix over F , and we can work out det det M, which will b e an R F R n n m m m n n m mn mn elementofF . On the other hand, since R F ,wehave M 2 R F = F , so we can work out det M, which will also b e an element of F . Our main conclusion is F that these two calculations give the same result: n n Theorem 1. Let R be a commutative subring of F , where F is a eld or a commu- m m tative ring, and let M 2 R . Then det M = det det M: 1 F F R 1 ! A B For example, if M = where A, B, C, D are n n matrices over F which all C D commute with each other, then Theorem 1says det M = det AD BC: 2 F F Theorem 1 will be proved later. First, in section 2 we shall restrict attention to the case m = 2 and give some preliminary and familiar results ab out determinants of blo ck diagonal and blo ck triangular matrices which, as a by-pro duct, yield a pro of by blo ck matrix techniques of the multiplicative prop erty of determinants. In section 3 we shall prove something a little more general than Theorem 1 in the case m =2; and Theorem 1 itself, for general m, will b e proved in section 4. 2 The multiplicative prop erty ! A B n n 2n 2n Let M = , where A, B, C, D 2 F , so that M 2 F . As a rst case, C D ! A O , a blo ck-diagonal supp ose B = C = O, the n n zero matrix, so that M = O D matrix. It isawell-known fact that ! A O det = det A det D: 3 F F F O D The keen-eyed reader will notice immediately that, since det A det D = det AD; 4 F F F equation 3 is just 2 in the sp ecial case where B = C = O. However, we p ostp one this step, b ecause with care we can obtain a proof of the multiplicative prop erty 4 as aby-pro duct of the main argument. One way of proving 3 is to use the Laplace expansion of det M by the rst n rows, F which gives the result immediately. A more elementary pro of runs thus: generalize to the case where A is r r but D is still n n. The result is now obvious if r = 1, by expanding by the rst row. So use induction on r , expanding by the rst row to p erform the inductive step. Details are left to the reader. This result still holds if we know only that B = O, and the pro of is exactly the same: we obtain ! A O det = det A det D: 5 F F F C D By taking transp oses, or by rep eating the pro of using columns instead of rows, we also obtain the result when C = O, namely, ! A B = det A det D: 6 det F F F O D 2 In order to prove 4 we need to assume something ab out determinants, and we shall assume that adding a multiple of one row resp ectively, column to another row resp ec- tively, column of a matrix do es not alter its determinant. Since multiplying a matrix on the left resp ectively, right by a unitriangular matrix corresp onds to p erforming a number of such op erations on the rows resp ectively, columns, it do es not alter the de- terminant. A uni triangular matrix is a triangular matrix with all diagonal entries equal to 1. We shall also assume that det I =1, where I is the n n identity matrix. So F n n now observe that ! ! ! ! ! I I A B C D I I I O n n n n n = ; 7 C D A B I I O I O I n n n n ! ! C D A B , since the rst three matrices on the left = det whence det F F A B C D of 7 are unitriangular. From 5 and 6 it follows from this that ! ! O B A B : 8 = det C det B = det det F F F F C D C O But also ! ! ! A AD A O I D n = : I O I D O I n n n Here the second matrix on the left is unitriangular, so taking determinants and using 5 and the rst part of 8, we have det A det D = det I det AD; F F F n F n n and since det I =1, the multiplicativelaw 4 for determinants in F follows. F n 3 Determinants of 2 2 blo ck matrices Since we now know that det A det D = det AD, then also det C det B = F F F F F det B det C = det BC = det BC. From 5, 6 and 8, we obtain: F F F F ! A B Lemma 2. If M = , then C D det M = det AD BC 9 F F whenever at least one of the blo cks A, B, C, D is equal to O. Compare this with 2. We shall now try to generalize somewhat. Supp ose the blo cks C and D commute, that is, CD = DC. Then ! ! ! ! AD BC B AD BC B D O A B : 10 = = O D CD DC D C I C D n 3 n n 2n 2n We proved 4 in F for all n, so we can apply it in F to get, via 5 and 6, det M det D = det AD BC det D; F F F F and so det M det AD BC det D =0: 11 F F F Now if det D is not zero or, in the case where F is a ring rather than a eld, if det D is F F not a divisor of zero, then 9 follows immediately from 11; but we do not actually need this extra assumption, as we shall now show. Adjoin an indeterminate x to F , and work in the p olynomial ring F [x]. This is another commutative ring, and a typical elementis r r 1 a p olynomial a x + a x + :::+ a x + a , where a 2 F , all i, and addition, subtraction 0 1 r 1 r i and multiplication of p olynomials is done in the obvious way.
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