One of the biggest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude. But we need proof for all natural numbers.
Answers
Answer:
.
Indeed, this proof is perfectly valid for matrices over G.
PROBLEMS
1. Prove that the determinant of a triangular matrix (i.e., a matrix which is either upper-triangular or a lower triangular) is the product of its diagonal entries.
2. Generalize the previous problem to showing that the determinant of a block-triangular matrix is the product of the determinants of its diagonal blocks.
3. For a 2� 2 matrix A over a field F , show the equation AX = I, where X is a 2� 2 matrix, has a solution and that it is unique iff det A � 0; and thereby obtain a formula for A-1.
4. Let C be invertible such that C-1AC = U is an n� n upper triangular matrix. Prove that det A = u11 u11 � unn.
5. Let A be a 2x2 matrix over a field F. Show that there holds det (cI-A) = c2, for each c Î F iff A2 = 0.
6. If a function D : Fn� n ® F satisfies D(AB) = D(A)D(B) for all A, B, prove that either D(A) = 0 for all A, or D(I) = 1. Give an example of a function D of a latter type which is not the same as the determinant function.
7. If D : ℝ2� 2 ® ℝ satisfies D(AB) = D(A)D(B) for all A, B and if , show that
(a) ; (b) = -1, D(0) = 0; (c) D(A) = 0, whenever |A| = 0.
8. Let G denote a commutative ring with identity and G n� n the set of all n� n matrices over G . A map D : G n� n ® G is said to be n-linear if D(A) is linear with respect to each of the n-rows of A while the other n-1 rows are kept fixed. D is called alternating if D(A) = 0 whenever any two rows of A are identical. Show that if D is n-linear and alternating and à is a matrix obtained by interchanging any two rows in A, then D(Ã) = - D(A).
9. If G is a commutative ring with identity such that g + g = 0, g Î G implies g = 0, and if a map D : Gn� n ® G has the property D(Ã) = - D(A), where à is obtained from A by interchanging any two of its rows, show that D is alternating.
10. If G is a commutative ring with identity 1, D : Gn� n ® G is (i) n-linear, (ii) alternating and (iii) satisfies D(I) = 1, show that D coincides with the determinant function on Gn� n.
11. Which of the following D's are 3-linear functions on ℝ3� 3?
(a) D(A) = a11 a22 a33;
(b) D(A) = (a11 + a22 + a33)(a11 + a22 + a33 );
(c) D(A) = a11 + a22 + a33;
(d) D(A) = a11 a21 a31 + a12 a22 a32 + a13 a23 a33;
(e) D(A) = 0;
(f) D(A) = -1.
12. For an n� n matrix A over F , verify that D(A) = A(j1,k1)A(j2,k2) ... A(jn,kn). is n-linear if and only if the integers j1, ... , jn are distinct.
13. If G is a commutative ring with identity, verify that the determinant function on the set of 2� 2 matrices over K is alternating and 2-linear as a function of the rows as well as the columns of A. Does the result generalize to n� n matrices?
14. If G is a commutative ring with identity and D is an alternating n-linear funciton on Gn� n, show that (a) D(A) = 0, if one of the rows of A is 0 and (b) D(B) = D(A), if B is obtained from A by adding a scalar multiple of one of its rows to another.
15. For A Î F3� 3, if
,
prove that if a31 c1 + a32 c2 + a33 c=3 = 0 and (c1, c2, c3)� � 0, then rank A = 2 and that (c1, c2, c3)� is a basis for the solution space of the system of equations Ax = 0. Think about an appropriate generalization in the n� n case.
16. Let G be a commutative ring with identity, and let D be an alternating n-linear funciton on Gn� n. Show that D(A) = (det A)D(I) for all A and deduce that det (AB) = (det A)(det B) for any A, B Î Gn� n.
17. If F is a field, the set of all matrices of the form f(A), where f is a polynomial over F and A Î F2� 2 , is a commutative ring
Answer:
please go to Google buddy
but follow me here and thanks my all answer