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Assignment 3 Let (5 pt.) A= (1 14 -1) -3) (a) Give matrices D and N...
Please Answer All The Question 3. An n x n matrix N is called nilpotent if...
Please Answer All The
Question
3. An n x n matrix N is called nilpotent if Nd = 0 for some d. (a) Show that if N is nilpotent, then N" = 0. (b) Show that the dim ker N is the number of Jordan blocks in its Jordan canonical form (c) How many similarity classes of 5 x 5 nilpotent matrices are there?
Need help!! 1) Let A, B, C, and D be the matrices defined below. Compute the...
Need help!!
1) Let A, B, C, and D be the matrices defined below. Compute the matrix expressions when they are defined; if an expression is undefined, explain why. [2 0-1] [7 -5 A= .B -5 -4 1 C- ,D= (-5 3] [I -3 a) AB b) CD c) DB d) 3C-D e) A+ 2B 2) Let A and B be the matrices defined below. 4 -2 3) A=-3 0, B= 3 5 a) Compute AB using the definition of...
3. Let Y ~ N(aln, σ21n) and matrices B and A be such that BY and...
3. Let Y ~ N(aln, σ21n) and matrices B and A be such that BY and (n-1)s-YAY (a) Show that B = n-11, and A = 1-n-J where I is the identity matrix and J is the matrix of all ones (b) Show that A is idempotent. (c) Show that tr(A)- rank(A). ( d ) Compute AB .
3. Let Un (R) be the subgroup of GLn(R) consisting of upper triangular matrices and let...
3. Let Un (R) be the subgroup of GLn(R) consisting of upper triangular matrices and let Dn(R) be the subgroup of GLn(R) consisting of diagonal matrices. (a) Show that \ : Un(R) + Dn(R), A + diag(a11, ..., ann) is a homomorphism of groups. Find the kernel and image of y. (b) Let Zn(R) = {aIn : a E R*} be the subgroup of Dn(R) consisting of scalar matrices. Determine x-1(Zn(R)). Justify your answers.
(1 pt) Determine which of the formulas hold for all invertible n x n matrices A...
(1 pt) Determine which of the formulas hold for all invertible n x n matrices A and B 21, O B, (A + B)2-A2 + B2 + 2AB 11212 D. A+ B is invertible E. ABAB F. 9A is invertible (1 pt) Solve for X. 4 -2 -9-8 7J-L-6-3 8 -3
12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that...
I
will give a rate! please show work clearly! thanks!
12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.
12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.
Let A, B, C, and D be matrices with the following sizes: A, 5×3 B, 3×2...
Let A, B, C, and D be matrices with the following sizes: A, 5×3 B, 3×2 C, 3×5 D, 1×3 Which of the following matrix operations are defined? i) AB (ii) A + 1 4 C (iii) DC
6. a. Show that if N E C(H) is nilpotent then ơr(N) 0, (use 6d fron last assignment). b. List the...
please answer #6 a and b, my 6d from previous assignment is
shown in 2nd picture
6. a. Show that if N E C(H) is nilpotent then ơr(N) 0, (use 6d fron last assignment). b. List the similarity classes of the (nonzero) nipotent linear maps of a 5- dimensional vector space overE i.e., give a representative matrix in each class) d) N is nilpotent z-3n such that Nn , D T-Nisinvertible wit'h inverse ANAN .
6. a. Show that if...
1. Let R be a commutative ring with identity and let u e R be nilpotent...
1. Let R be a commutative ring with identity and let u e R be nilpotent elements a) (3 pt) Show that x + y and xy are nilpotent elements. b) (3 pt) Show that if u is a unit of R and t is nilpotent, then u is a umit. ) 3 pt) Show that if R is not commutative, neither of the above necessarily holds (r t y is not necessarily nilpotent and u 4- r is not...
1. (10 points) Let A and B be 3 x 3 matrices, with det A =...
1. (10 points) Let A and B be 3 x 3 matrices, with det A = -3 and det B = 2. Compute (a) det AB (6) det B4 (c) det 3B (d) det A"B" AT (e) det B-AB
Please Answer All The
Question
3. An n x n matrix N is called nilpotent if Nd = 0 for some d. (a) Show that if N is nilpotent, then N" = 0. (b) Show that the dim ker N is the number of Jordan blocks in its Jordan canonical form (c) How many similarity classes of 5 x 5 nilpotent matrices are there?
Need help!!
1) Let A, B, C, and D be the matrices defined below. Compute the matrix expressions when they are defined; if an expression is undefined, explain why. [2 0-1] [7 -5 A= .B -5 -4 1 C- ,D= (-5 3] [I -3 a) AB b) CD c) DB d) 3C-D e) A+ 2B 2) Let A and B be the matrices defined below. 4 -2 3) A=-3 0, B= 3 5 a) Compute AB using the definition of...
3. Let Y ~ N(aln, σ21n) and matrices B and A be such that BY and (n-1)s-YAY (a) Show that B = n-11, and A = 1-n-J where I is the identity matrix and J is the matrix of all ones (b) Show that A is idempotent. (c) Show that tr(A)- rank(A). ( d ) Compute AB .
3. Let Un (R) be the subgroup of GLn(R) consisting of upper triangular matrices and let Dn(R) be the subgroup of GLn(R) consisting of diagonal matrices. (a) Show that \ : Un(R) + Dn(R), A + diag(a11, ..., ann) is a homomorphism of groups. Find the kernel and image of y. (b) Let Zn(R) = {aIn : a E R*} be the subgroup of Dn(R) consisting of scalar matrices. Determine x-1(Zn(R)). Justify your answers.
(1 pt) Determine which of the formulas hold for all invertible n x n matrices A and B 21, O B, (A + B)2-A2 + B2 + 2AB 11212 D. A+ B is invertible E. ABAB F. 9A is invertible (1 pt) Solve for X. 4 -2 -9-8 7J-L-6-3 8 -3
I
will give a rate! please show work clearly! thanks!
12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.
12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to A.
please answer #6 a and b, my 6d from previous assignment is
shown in 2nd picture
6. a. Show that if N E C(H) is nilpotent then ơr(N) 0, (use 6d fron last assignment). b. List the similarity classes of the (nonzero) nipotent linear maps of a 5- dimensional vector space overE i.e., give a representative matrix in each class) d) N is nilpotent z-3n such that Nn , D T-Nisinvertible wit'h inverse ANAN .
6. a. Show that if...
1. Let R be a commutative ring with identity and let u e R be nilpotent elements a) (3 pt) Show that x + y and xy are nilpotent elements. b) (3 pt) Show that if u is a unit of R and t is nilpotent, then u is a umit. ) 3 pt) Show that if R is not commutative, neither of the above necessarily holds (r t y is not necessarily nilpotent and u 4- r is not...
1. (10 points) Let A and B be 3 x 3 matrices, with det A = -3 and det B = 2. Compute (a) det AB (6) det B4 (c) det 3B (d) det A"B" AT (e) det B-AB
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