Problem 3. Give the definitions of an invertible square matrix and of the inverse of Let A be a s...
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)...
Give an example of a matrix A that has a left inverse but does not have a right inverse. (If BA = I then B is a left inverse of A.) 2. Give an example of a matrix A that has a right inverse but does not have a left inverse. (If AB = I then B is a right inverse of A.) Let V and W be vector spaces. If T 2 L(V;W) is invertible then T is called...
Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant. Let A and B be square matrices and P be an invertible matrix. If A- PBP-,show that A and B have the same determinant.
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to 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...
Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent. Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent.
「 : / (2) Let A- be an arbitrary 2 x 2 matrix. (a) If A is invertible, perform row operations to determine a row echelon form of A. (Hint: You may need to consider different cases, e.g., when a-0 and when a f 0.) (b) Under certain conditions, we can row reduce [A | 2 to [| B] where d -b ad- be-a Use the row echelon form of A from part (a) to find conditions under which the...
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
An invertible square matrix A satisfies A^3 +3A^2 −25A+21I = O, where I and O are the identity and zero matrices, respectively. Find the inverse of A^2
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.