Exercise 2.1.48 Show that if A-1 exists for an n × n matrix, then it is...
Show that, given an m x n matrix A, there exists a unique matrix B satisfying the following properties: 1. AB and BA are symmetric. 2. ABA = A 3. BAB = B
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n. A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
If Matrix A, r(A)=n, prove that r(AB)=r(B), for any B nxp, and show that for any invertible mxm matrix P, there exists Q mxn with full rank such that A=PQ
6. Show that if A is an n x n symmetric matrix and B is an n x m matrix. Show that BT AB, BT B and BBT are symmetric matrices (10 pts)
(a) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any le R, we can write A = XI + (A - XI) (b) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn.n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M e Mn,n(R) such that M&V....
Problem 4.26 194 Chap. 4 Duality theory Exercise 4.26 Let A be a given matrix. Show that exactly one of the following alternatives must hold. (a) There exists some x 0 such that Ax = 0, x > 0. (b) There exists some p such that p'A>0'. Exercise 4.27 Let A be a given matrix. Show that the following two state- ments are equivalent. (a) Every vector such that Ax > 0 and x > 0 rnust satisfy x1 =...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes. Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes.