By O. A. Nielsen
ISBN-10: 0824769716
ISBN-13: 9780824769710
Publication by means of Nielsen, O. A.
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3,' 4. un for Solution. Let -4 be an n x n matrix. The expansion of det(^4) by row 1 is n det(4) = V a-. 4.. . Now A. is an (n - 1) - square determinant, so % =1 we need u * operations to compute it. Hence to form all the a1 . A, • 71—1 calls for 11 n (u _. + 1) n numbers to get + 271-1. Obviously operations. det(i4). ^ = 0; Hence 4w3 + 7 = 63. Finally, we still have to add up these u = n (u , + l) + 7 i - l = using the formula, we obtain (which we can also see directly). 3 6 1. For the matrices identity 3 A = and 4 -3 det(i4)det(B).
Now A. is an (n - 1) - square determinant, so % =1 we need u * operations to compute it. Hence to form all the a1 . A, • 71—1 calls for 11 n (u _. + 1) n numbers to get + 271-1. Obviously operations. det(i4). ^ = 0; Hence 4w3 + 7 = 63. Finally, we still have to add up these u = n (u , + l) + 7 i - l = using the formula, we obtain (which we can also see directly). 3 6 1. For the matrices identity 3 A = and 4 -3 det(i4)det(B). det(i4B) = 2 5 1 4 7 verify the B = Solution. Evaluate both sides of the equation to get the same answer 180.
Apply the row operations R^ - i ^ and R^ - R^ to resulting determinant rows 2 and 3 both consist entirely of Dn = 0. (Note however that 10. LO. D1 = 2 and Z> ; l's. ) Let w w un denote the number of additions,' subtractions subtrad >ns a: and ' n r( needed to to evaluate evaluate an an n x n7i determinant by using u ing re itiplii cations needed nultipli row ultipli multiplications needed to evaluate an ove tr Prove that Prove that expansion. nsion. iansion. :pansion. expansion. late :ulate Llculate »o1/»iilo + o calculate Prove that ni u u un = = u n x n +,, + nu nu n-l + 2n 2n -- determinant by using row 1.
Direct Integral Theory by O. A. Nielsen
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