a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2.
b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
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a) Let I be the n x n identity matrix and let O be the n × n zero matrix
Problem 2 Let A be an n x n matrix which is not 0 but A-0...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is inv...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
(1 point) A matrix A is said to be similar to a matrix B if there...
(1 point) A matrix A is said to be similar to a matrix B if there is an invertible matrix P such that B = PAP 1 Let A1, A2, and A3 be 3 x 3 matrices Prove that if A1 is similar to A2 and A2 is similar to A3, then A similar to A. Proof: Since A1 is similar to A2, for some invertible matrix P for some invertible matrix Q Since A2 is similar to A3 for...
3.23 True or false. justify your answer 190 LINEAR TRANSFORMATIONS 3.22 Let A be a 4...
3.23 True or false. justify your answer
190 LINEAR TRANSFORMATIONS 3.22 Let A be a 4 x 3 matrix and B a 3 x 4 matrix. Then AB cannot be in 3.23 Suppose that A is an invertible matrix and B is any matrix for which BA i 3.24 Suppose that A is an invertible matrix and B is any matrix for which AB is 3.25 Suppose that A and B are nxn matrices such that AB is invertible. Then...
Help on Questions 1-3 Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
4. Let A be an n x n matrix. Define the trace of A by the...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are simil...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
In the next exercises, we consider square n X n matrices; I is the identity matrix...
In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in...
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.
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and...
a. Let B be an n x n Orthogonal matrix, that is B^-1 = B^T, and let A be an n x n skew-symmetric matrix. Simplify A(A^2(BA)^-1)^T b. Let A be a square matrix such that A^3 = 0. A is then called a nilpotent matrix. Define another matrix B by the expression B = I - A; Show that B is invertible and that its inverse is I + A + A^2 c. Let B = (-2,0,0 ; 0,0,0...
Problem 2 Let A be an n x n matrix which is not 0 but A-0 Let I be the identity matrix. a) (10 Points) Show that A is not diagonalizable. b) (5 Points) Show that A is not invertible. e) (5 Points) Show that I-A is invertible and find its inverse.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
(1 point) A matrix A is said to be similar to a matrix B if there is an invertible matrix P such that B = PAP 1 Let A1, A2, and A3 be 3 x 3 matrices Prove that if A1 is similar to A2 and A2 is similar to A3, then A similar to A. Proof: Since A1 is similar to A2, for some invertible matrix P for some invertible matrix Q Since A2 is similar to A3 for...
3.23 True or false. justify your answer
190 LINEAR TRANSFORMATIONS 3.22 Let A be a 4 x 3 matrix and B a 3 x 4 matrix. Then AB cannot be in 3.23 Suppose that A is an invertible matrix and B is any matrix for which BA i 3.24 Suppose that A is an invertible matrix and B is any matrix for which AB is 3.25 Suppose that A and B are nxn matrices such that AB is invertible. Then...
Help on Questions 1-3
Math 311 Orthogonal & Symmetric Matrix Proofs 1. Let the n x n matrices A and B be orthogonal. Prove that the sum A + B is orthogonal, or provide counterexample to show it isn't 2. Let the n x n matrix A be orthogonal. Prove A is invertible and the inverse A-1 is orthogonal, or provide a counterexample to show it isn't. 3. Suppose A is an n x n matrix. Prove that A +...
4. Let A be an n x n matrix. Define the trace of A by the formula tr(A) = 2 . In other words, the trace of a matrix is the sum of the diagonal entries of the matrix. It is known that for two n x n matrices A and B, the trace has the property that tr(AB) = tr(BA). Each of the following holds more generally, for n x n matrices A and B, but in the interest...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
In the next exercises, we consider square n X n matrices; I is the identity matrix (In MATLAB eye (n) gives a square n by n matrix). If ex is the column unit vector which components are all O's except the kth component which is equal to 1, i.e., 이".e1 = 101 0 then the identity matrix I is such that: 10 000 ei = [100 01T, , el 1000 11T. 0 0 T' 0 0 1 To generate in...
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.
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