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1. To prove the theorem in detail. Theorem: det A for any n X n-matrix A can be computed by a cof...
Let A be an n×n matrix. Mark each statement as true or false. Justify each answer. a. An n×n determinant is define...
Let A be an n×n matrix. Mark each statement as true or false. Justify each answer. a. An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices. b. The (i,j)-cofactor of a matrix A is the matrix obtained by deleting from A its I’th row and j’th column. a. Choose the correct answer below. A. The statement is false. Although determinants of (n−1)×(n−1)submatrices can be used to find n×n determinants,they are not involved in the definition of n×n determinants. B....
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE r=1(no induction required, just use the definition of the determ...
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE
r=1(no induction required, just use the definition of the
determinants)
Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
Prove the following lemma. Let B be an n ✕ n matrix and let E be...
Prove the following lemma.
Let B be an n ✕ n
matrix and let E be an n ✕ n
elementary matrix. Then det(EB) = det(E)
det(B)
1. Write the proof and submit as a free response. (Submit a file
with a maximum size of 1 MB.)
2. Which of the following could begin a direct proof of the
statement?
If E interchanges two rows, then det(E) = 1 by
Theorem 4.4. Also, EB is the same as B but...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A us...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
Compute the determinant of the following n x n matrix: [ 2-1 -1 2-1 -1 2...
Compute the determinant of the following n x n matrix: [ 2-1 -1 2-1 -1 2 2 -1 -1 2 (All "missing" entries are 0.) This is a nice exercise in mathematical induction. To do this problem, first try computing a few specific cases, then make a conjecture about the general nxn determinant. Then prove that your conjecture is correct by induction. (Actually, you will use strong induction, where you assume that the determinant in the nxn case follows from...
LU be an LU factorization of matrix A e Fn×n computed by the Gaussian elimination Let...
LU be an LU factorization of matrix A e Fn×n computed by the Gaussian elimination Let PA with partial pivoting (GEPP). Let us denote Prove that (a) leyl 1, for all i >j S 2-1 maxij laijl You may assume P-1, i.e., in each step of the Gaussian elimination process the absolute value of the diagonal entry is already the largest among those of the entries below the diagonal entry on the same column You may prove the results with...
Explain all parts of question 1 and question 2 in detail 1. Consider the matrix In...
Explain all parts of question 1 and question 2 in detail
1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. E...
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. Every eigenvalue of A lies in at least one of the m circular disks in the complex plane with centers au and radii Σ (41. Moreover, if n of these disks forrn a connected domain that is disjoint from the other m-n disks, then there are precisely n eigenvalues of A within this domain. Prove the first part of Gerschgorin's theorem. (Hint: Let...
13 please 8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3...
13 please
8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE
r=1(no induction required, just use the definition of the
determinants)
Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
Prove the following lemma.
Let B be an n ✕ n
matrix and let E be an n ✕ n
elementary matrix. Then det(EB) = det(E)
det(B)
1. Write the proof and submit as a free response. (Submit a file
with a maximum size of 1 MB.)
2. Which of the following could begin a direct proof of the
statement?
If E interchanges two rows, then det(E) = 1 by
Theorem 4.4. Also, EB is the same as B but...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
Compute the determinant of the following n x n matrix: [ 2-1 -1 2-1 -1 2 2 -1 -1 2 (All "missing" entries are 0.) This is a nice exercise in mathematical induction. To do this problem, first try computing a few specific cases, then make a conjecture about the general nxn determinant. Then prove that your conjecture is correct by induction. (Actually, you will use strong induction, where you assume that the determinant in the nxn case follows from...
LU be an LU factorization of matrix A e Fn×n computed by the Gaussian elimination Let PA with partial pivoting (GEPP). Let us denote Prove that (a) leyl 1, for all i >j S 2-1 maxij laijl You may assume P-1, i.e., in each step of the Gaussian elimination process the absolute value of the diagonal entry is already the largest among those of the entries below the diagonal entry on the same column You may prove the results with...
Explain all parts of question 1 and question 2 in detail
1. Consider the matrix In + Inn, which has every diagonal entry equal to 2 and every off-diagonal entry equal to 1. (a) Compute det(In + Inn) for each of n = 1,2,3. (b) For n = 4, we have 2 1 1 1 1 2 1 1 1 1 2 1 111 2 2 1 1 1 -1 1 0 0 -1 0 1 0 -1 0 0...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Here is Gerschgorin's theorem, which holds for any m x m matrix A, symmetric or nonsym- metric. Every eigenvalue of A lies in at least one of the m circular disks in the complex plane with centers au and radii Σ (41. Moreover, if n of these disks forrn a connected domain that is disjoint from the other m-n disks, then there are precisely n eigenvalues of A within this domain. Prove the first part of Gerschgorin's theorem. (Hint: Let...
13 please
8. b. -2 3 0 0 0 0 -1 2 0 0-4 0 3 0-2 0 3 0 0 -2 0 3 0 4 o0-1 6 0 0 1 o 2 6 0 0 -1 6 10. For any positive integer k, prove that det(4t) - de(A)*. 11. Prove that if A is invertible, then den(A-1)- I/der(A) - det(4)- 12. We know in general that A-B丰B-A for two n x n matrices. However, prove that: det(A . B)-det(B...
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