DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS

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1 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS DEANNA LEGGETT, JOHN PERRY, AND EVE TORRENCE ABSTRACT We consider the problem of zeroes appearing in the interior of a matrix when computing determinants with Dodgson s method We introduce a modification of the method that resolves the problem in many important and interesting cases We show that this modification, called the double-crossing method, generalizes Dodsgon s method 1 INTRODUCTION Algebra students learn the familiar formula to compute the determinant of a 2 2 matrix: a b ad bc c d A similar pattern exists for 3 3 matrices, but larger matrices require the student to learn something a little more complicated Most students usually learn to compute determinants by expansion of minors, first developed by Lagrange Some students learn to compute matrices by triangularizing the matrix Both methods are effective, and triangularization is efficient, but students tend to dislike them In 1866, the Rev Charles Lutwidge Dodgson, better known as Lewis Carrol of Alice in Wonderland fame, developed a conceptually simple method to compute determinants [2] Dodgson s method iterates the familiar 2 2 formula Example 1 Given the matrix A we set A 3 A To compute A 2, compute the determinants of the two-by two submatrices of A 2 : 1 0 A To compute A 1, repeat the computation for A 2, but divide by the interior of A 3 : A , Now go back and compute A using your favorite method expansion of cofactors, triangularization, etc What did you get? Dodgson s method is quick and conceptually simple In general, we can describe the method in the following way: Let A n be the n n matrix given For each k n 1, n 2,, 1: Let B k be the k k matrix of determinants of 2 2 matrices of A k+1 ; 1

2 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 2 Let A k be the k k matrix whose i, j-th element is the i, j-th element of B divided by the i, j-th element of the interior of A k+2 If k + 2 > n, do not perform the division The singleton element of A 1 is the determinant of A Dodgon s method is an example of a condensation method to compute determinants; each iterate A k is a condensation of the previous iterate A k 1 Similar methods appear in [1] Unfortunately, Dodgson s method has a huge drawback, and the attentive eye will have noticed it already: division What s so bad about division? Example 2 Swap the first two rows of the matrix of Example 1 to obtain M , The determinant of this new matrix is -3 What happens when we use Dodgson s method to compute M 1? Did you catch it? To compute M 1, you will take the determinant of M 2, and divide by the interior element of M 3 But M 3 2,2 0! So Dodgson s method fails to compute the determinant of a matrix A whenever a zero appears in the interior of A k for any k > 2 You can swap rows to make the method work in certain circumstances, moving zeroes out of the interior and onto the exterior; swapping the top two rows of M in Example 2 gets us back to A of example 1, for which Dodgson s method worked fine However, there are two drawbacks to this method First, swapping rows may well introduce other zeroes into the matrix, and it isn t easy to predict this from the outset Second, swapping rows simply won t work for some matrices Example 3 It is impossible to swap rows in such a way that allows us to compute N 2 in Dodgson s method if N Will a different repair of Dodgson s method work? Yes! By re-examining its relationship to a theorem of Jacobi, we found a modification that, by the end of this paper, will allow us to compute the determinant of N This modification succeeds at computing the determinant not only when one zero appears in the interior, but even when many zeroes appear in the interior! 2 THE FIX: A DOUBLE-CROSSING METHOD We describe the fix in this section The reader will see quickly why we call it a double-crossing method; an explanation of why it fixes Dodsgon s method most of the time appears in Section 4, after the explanation in Section 3 of why Dodgson s method works Example 4 Recall M from Example 2 The first two condensations in Dodgson s method are 21 M and M We encounter trouble when computing M 1, because of the zero in M 3 Above that interior zero element is a non-zero element, M 3 1,2 3 The fix allows us to divide by this element instead How? Re-compute M 2 in a slightly different manner Cross out the first row and second column of M 3 the ones containing 3 You are left with the 2 2 complementary matrix M

3 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 3 Consider the matrix M of adjoints of M 3 that correspond to the elements of M ; that is, 3 1 M How did we obtain the matrices within each determinant of M? The adjoint of an element is computed by again crossing out the row and column containing that element, and taking the determinant of what s left The upper left element of M is 1; it came from row 2, column 1 of M 3 Cross out row 2 and column 1 of M 3, and you find the matrix in the upper left corner of M Similarly, the lower left element of M is 0; it came from row 3, column 1 of M 3 Cross out row 3 and column 1 of M 3, and you find the matrix in the lower left corner of M Moving on, do the following Put 22 M 2 M conclude by dividing M 2 by the non-zero element of M 3 that we identified earlier: M 2 M As noted in example 2, det M 3 Notice, by the way, that the bottom row of M 2 in line 22 contains the same values as the top row of M 2 in line 21, only swapped We will say more about this later The fix preserves the spirit of Dodgson s method, inasmuch as we computed all determinants by condensing 2 2 matrices The example shows why we call it the double-crossing method: we found a non-zero element adjacent to the zero element; we crossed out its row and column, obtaining the complementary matrix M ; for each element in M, we again cross out the row and column of its location in M 3, and use the determinant of the remaining matrix to compute the element of M 2 When a zero appears in an intermediate matrix A k where n > k > 2, we cross out rows and columns that contain a submatrix of A n Let s look at a matrix where a zero does not appear in A, but does appear in an intermediate matrix Example 5 Let A ; Using Dodgson s method, we compute A 5, A 4, and then A We encounter a zero in the interior! Above it is a non-zero element, 2 Notice that it lies in the top row and central column of A 3 To compute A 2, do the following: cross out the top three rows and central three columns of A 5 the original matrix; this gives us a 2 2 complementary matrix A ;

4 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 4 determine the adjoints of the elements of A in A 5 ; these adjoints give you the matrix A 2 Following these instructions, we have A 0 0 and A We compute the determinants for A 2 using Dodgson s method and again, the bottom two values turn out to have values that you would obtain in the two top rows by ordinary condensation of A 3, while the others are different: A We now compute A 1 by dividing the determinant of A 2 by the non-zero entry of A 3 that we identified above: -2 We obtain A , 2 and in fact A 4 So far we ve been using a special case of the double-crossing method Let s describe it in a theorem Theorem 6 Double-crossing method, special case Let A be an n n matrix Suppose that we try to evaluate A using Dodgson s method, but we encounter a zero in the interior of A k, say in row i and column j of A k If the element α in row i 1 and column j of A k is non-zero, then we compute A k+2 as usual, with the following exception for the element in row i 1 and column j 1: write l n k; identify the l + 3 l + 3 submatrix B whose upper left corner is the element in row i l and column j l of A n ; identify the complementary matrix A by crossing out the l + 1 l + 1 submatrix of B whose upper left corner is the element in row 1 and column 2 of B; compute the matrix A of adjoints of A in B; compute the element in row i 1 and column j 1 of A k+2 by dividing the determinant of A by α We can use Dodgson s method to compute the intermediate determinants A proof of correctness follows after we explain why Dodgson s original method works correctly One can generalize Theorem 6 so that the non-zero element appears immediately above, below, left, right, or catty-corner to the zero; that is, the non-zero element is adjacent to the zero If the zero appears in a 3 3 block of zeroes, then the double-crossing method will not repair Dodgson s method Before proceeding to the next section, we encourage the reader to go back and examine how Theorem 6 describes what we did in the examples of this section We conclude by applying the double-crossing method to compute the deterninant N from Example 3 Example 7 Recall from Example 3 N

5 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 5 Let N 4 N; we have two zero elements in the interior, at i, j 2, 3 and at i, j 3, 2 As described in Theorem 6, these correspond to elements 1, 2 and 2, 1 of N 2 The two other elements of N 2 can be computed as usual: N We use the theorem to compute the other two elements of N 2 N 2 1?? 2 Case 1 i, j 2, 3 The interior zero appears in the original matrix, so l 0 We choose the 3 3 submatrix whose upper left corner is the element in row 1, column 2 of N 4 : B Cross out the 1 1 submatrix in row 1, column 2 of B to obtain the complementary matrix A Compute the matrix of adjoints 1 0 A Thus N 2 1 0? Case 2 i, j 3, 2 The interior zero appears in the original matrix, so l 0 We choose the 3 3 submatrix whose upper left corner is the element in row 2, column 1 of N 4 : B Cross out the 1 1 submatrix in row 1, column 2 of B to obtain the complementary matrix A Compute the matrix of adjoints 1 0 A Thus N We can now conclude by computing N

6 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 6 In fact, the determinant of N is -1 3 JACOBI S THEOREM We now turn to an explanation of why Dodgson s method works in the first place In general, A R m n will denote a matrix, and A its adjugate, also known as its adjoint Both Dodgson s method and the fix make use of submatrices of a matrix, which for a given matrix A we will denote in the following fashion: A ij,kl A i,kl A ij,k rows i, i + 1,, j and columns k, k + 1,, l of A row i and columns k, k + 1,, l of A rows i, i + 1,, j and column k of A Example 8 Let A be the second matrix given in the introduction; that is, A Then A 12, and A 3, Dodgson s method relies on an elegant theorem of Jacobi Theorem Jacobi s Theorem Let Then A be an n n matrix; M an m m minor of A, where m < n, M the corresponding m m minor of A, and M the complementary n m n m minor of A det M det A m 1 det M We will not prove Jacobi s Theorem; a recent proof appears in [2] We ll use a generic 4 4 matrix to illustrate Dodgson s method and compare with various applications of Jacobi s Theorem Example 9 Consider a general 4 4 matrix A; we show how Dodgson s method applied Jacobi s Theorem The first condensation is A 12,12 A 12,23 A 12,34 A 3 A 23,12 A 23,23 A 23,34 A 34,12 A 34,23 A 34,34 and A 2 A 12,12 A 12,23 A 23,12 A 23,23 a 2,2 A 23,12 A 23,23 A 34,12 A 34,23 a 3,2 A 12,23 A 12,34 A 23,23 A 23,34 a 2,3 A 23,23 A 23,34 A 34,23 A 34,34 To see how A 2 is computed from Jacobi s method, consider the upper left 3 3 submatrix of A 4 Choose a1,1 a M 1,3 ; a 3,1 a 3,3 a 3,3

7 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 7 its complement in the 3 3 submatrix is M A 4 2,2 a 2,2 The 2 2 submatrix of A corresponding to the adjoints of M is a 2,2 a 2,3 M a 3,2 a 3,3 a 1,2 a 1,3 a 2,2 a 2,3 By Jacobi s Theorem, a 2,1 a 2,2 a 3,1 a 3,2 a 1,1 a 1,2 a 2,1 a 2,2 A23,23 A 23,12 A 12,23 A 12,12 det M det A 13, det M det M det M det A 13,13 A 12,12 A 12,23 A 23,12 A 23,23 det A a 13,13 2,2 This is precisely the element in the upper left corner of A 2 Look at the next and final condensation in Dodgson s method, A 1 Apply Jacobi s Theorem on A 4 ; A 2 A 23,23 A13,13 A 24,24 A 23,13 A 13,24 A 23,23 A12,12 A 12,23 A 23,12 A 23,23 This is precisely the singleton element of A 3! A A 13,13 A 24,24 A 23,13 A 13,24 A 23,23 We now state a specific case of Jacobi s Theorem on a submatrix B of A, where the 2 2 minor M consists of the corners of the submatrix In Lemma 10, i, j indicates the upper left corner of B, while indicates the row- and column-dimension of B Lemma 10 Let A be an n n matrix Let i, j {1,, n 1} and {1,, n max i, j + 1} If Ai+1i+ 2,j+1j+ 2 0, then Aii+ 2,jj+ 2 Ai+1i+ 1,j+1j+ 1 A ii+ 1,jj+ 1 A ii+ 2,j+1j+ 1 A i+1i+ 1,jj+ 2 Ai+1i+ 2,j+1j+ 2 Proof The proof is immediate from Jacobi s Theorem Figure 31 illustrates the 4 4 case considered in Example 9, and Figure 32, illustrates the arbitrary case Theorem 11 Let A be an n n matrix After k successful condensations, Dodgson s method produces the matrix A1k+1,1k+1 A1k+1,2k+2 A1k+1,n kn A n k A2k+2,1k+1 A2k+2,2k+2 A2k+2,n kn An kn,1k+1 An kn,2k+2 An kn,n kn whose entries are the determinants of the k + 1 k + 1 submatrices of A By successful iterations we mean that one never encounters division by zero

8 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 8 FIGURE 31 Diagram illustrating how Jacobi s Theorem is applied to Dodgson s method, given a 4 4 matrix see Example 9 Consider the generic 4 4 matrix from Example 9 The final condensation from Dodgson s method gave us A 1 A13,13 A 24,24 A 24,13 A 13,24 A 23,23 For Jacobi s Theorem, select the 2 2 minor A 1,1 A 1,4 M A 4,1 A 4,4 Its complementary minor is M A 23,23 Its adjugate is A 1,1 A 1,4 M A 4,1 A 4,4 where A 1,1 A 24,24, A 4,4 A 13,13, A 1,4 A 24,13, and A 4,1 A 13,24 By Jacobi s Theorem, det M det A 2 1 det M det M det M det A A 24,24 A 13,13 A 24,13 A 13,24 A 23,23 det A Notice that det A is the singleton element of A 1 Proof We proceed by induction on k Inductive Base: When k 1, the theorem is trivial: one condensation gives A 12,12 A 12,23 A 12,n 1n A n 1 A 23,12 A 2i+2,2i+2 A 23,n 1n A n 1n,12 A n in,2i+2 A n 1n,n 1n Inductive Hypothesis: Fix k Assume that for all l 1,, k, the lth condensation gives us A n l where for all 1 i, j n A n l i,j A ii+l,jj+l

9 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 9 FIGURE 32 Diagram illustrating how Jacobi s Theorem is applied to Dodgson s method Consider a submatrix of A, B A ii+ 1,jj+ 1 Select the 2 2 minor A i,j A i,j+ 1 M A i+ 1,j A i+ 1,j+ 1 Its complementary minor is M A i+1i+ 2,j+1j+ 2 The submatrix of B corresponding to the adjoints of M is where By Jacobi s Theorem, or 31 det B A i,j A i,j+ 1 M A i+ 1,j A i+ 1,j+ 1 A i,j Ai+1i+ 1,j+1j+ 1 A i,j+ 1 Ai+1i+ 1,jj+ 2 A i+ 1,j A ii+ 2,j+1j+ 1 A i+ 1,j+ 1 A ii+ 2,jj+ 2 A i,j det M det B 2 1 deg M A i,j+ 1 A i+ 1,j Ai+1i+ 2,j+1j+ 2 A i+ 1,j+ 1 In Example 9 we saw that with i j 1 and 3 the upper left element of A 2 had the form of 31 Inductive Step: Let i, j {1,, n k} The next condensation in Dodgson s method gives us A k+1 i,j Ak i,j A k i+1,j+1 Ak i+1,j Ak A k 1 i+1,j+1 i,j+1

10 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 10 From the inductive hypothesis, we can substitute Ai+1i+k+1,j+1j+k+1 A ii+k,jj+k Aii+k,j+1j+k+1 A k+1 A i+1i+k+1,jj+k i,j A i+1i+k,j+1j+k Using k + 2 in Lemma 10, this simplifies to A k+1 i,j A ii+k+1,jj+k+1 4 WHY THE FIX WORKS! The new workaround is based on Theorem 11 The goal in step i of the algorithm, is to compute each determinant A ki+k,ji+j Dodgson s method fails when the corresponding denominator from Lemma 10 is zero However, the fraction from Lemma 10 is not the only way to apply Jacobi s Theorem As long as some i 1 i 1 minor of A ki+k,ji+j has non-zero determinant, we can still recover, reusing most of the computations already performed, from Dodgson s method, calculating only a few new minors using the same approach as Dodgson s method For example, the proof of Theorem 6 can be summarized by applying Jacobi s Theorem with B in Theorem 6 standing in for A of Jacobi s theorem; the l + 1 l + 1 submatrix A of Theorem 6 standing in for M of Jacobi s Theorem; and the adjoints A of Theorem 6 standing in for M of Jacobi s Theorem Example 12 Recall N from Example 3 Dodgson s method would have us compute N 2 1,2 by choosing for Jacobi s Method N1,2 N M 1,4 M N N 3,2 N 2,3, 34 whence N 2 1,2 det N 13,24 det M det M but det M 0 The double-crossing fix allows us to choose instead N2,2 N M 2,4 N 3,2 N 3,4 M N 1,3 ; since N 1,3 1 we have N 2 1,2 det N 13,24 det M det M det M 1,1 det M 2,2 det M 1,2 det M 2, We conclude by showing how Jacobi s Theorem likewise justifies the fix of Example 5

11 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 11 FIGURE 41 Diagram for workaround for Example 12 The problem in computing N 2 1,2 was the zero in position N 2,3 Since N 2 1,2 det N 2,4,24 choose instead the minor N2,2 N M 2,4 N 3,2 N 3,4 whose complementary minor is M N 1,3 1 In this case the adjugate is N 1,3 N 1,4 M N 3,3 N 3,4 N 1,2 N 1,3 N 3,2 N 3,3 1 0 N 1,3 N 1,4 N 2,3 N 2,4 N 1,2 N 1, N 2,2 N 2, By Jacobi s Theorem, det M det B det M Notice that M is the minor created for N 2 1,2 by the double-crossing method in Example 7, we divided its determinant by 1, and obtained N 2 1,2 0 Example 13 Let A As before, we compute A 5, A 4, and then A which has a zero in the interior, corresponding to det A 24,24 This poses a problem for computing A 1 det A 5 Dodgson s method wants us to use for M the minor A1,1 A 1,5 A 5,1 A 5,5, whose complement is A 2,2 A 2,3 A 2,4 A 3,2 A 3,3 A 3,4 A 4,2 A 4,3 A 4,4

12 DETERMINANTS IN WONDERLAND: MODIFYING DODGSON S METHOD TO COMPUTE DETERMINANTS 12 The double-crossing method suggests instead to use A4,1 A M 4,5 A 5,1 A 5,5 whose complement is M the determinant of this matrix is N 3 1,2 to compute the adjoints of M in A 5 A 1,2 A 1,3 A 1,4 A 2,2 A 2,3 A 2,4 A 3,2 A 3,3 A 3, However, to compute N2 1,2 we need det M, which required us Of course, it is possible that both an element of the interior is zero and the element above it is zero The following theorem provides the promised generalization of Theorem 6, and its proof uses Jacobi s method in a manner similar to the proof of Theorem 6 at the beginning of this section Notice that if r s 0, then Theorem 14 generalizes to Dodgson s method: Indeed, if you stop to think about it, Dodgson s method is a double-crossing method! Theorem 14 Let A be an n n matrix Suppose that we try to evaluate A using Dodgson s method, but we encounter a zero in the interior of A k, say in row i and column j of A k Let r, s { 1, 0, 1} If the element α in row i + r and column j + s of A k is non-zero, then we compute A k+2 as usual, with the following exception for the element in row i 1 and column j 1: write l n k; identify the l + 3 l + 3 submatrix B whose upper left corner is the element in row i l and column j l of A n ; identify the complementary matrix A by crossing out the l + 1 l + 1 submatrix of B whose upper left corner is the element in row r + 2 and column s + 2 of B; compute the matrix A of adjoints of A in B; compute the element in row i 1 and column j 1 of A k+2 by dividing the determinant of A by α We can use Dodgson s method to compute the intermediate determinants REFERENCES [1] Alexander Aitken Determinants and Matrices Interscience Publishers, 1951 [2] Adrian Rice and Eve Torrence Shutting up like a telescope : Lewis Carroll s Curious Condensation Method for Evaluating Determinants, The College Mathematics Journal, address: dleggett0014@comcastnet UNIVERSITY OF SOUTHERN MISSISSIPPI address: johnperry@usmedu UNIVERSITY OF SOUTHERN MISSISSIPPI URL: address: etorrenc@rmcedu RANDOLPH-MACON COLLEGE ;

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